Sorry it took so long to post an answer. I've spent the past day researching this question, reading over two dozen PDFs (at least 1000 pages of material) and reading the Apollo 13 flight journal (which sadly, is incomplete).
It can't be emphasized enough just how important the guidance computers were to guidance and navigation. Every "normal" procedure used them directly. Their failure was considered, and the contingency plan relied on communication with mission control.
Similarly, communication with mission control was essential. The contingency plan for its failure was to use back-up programs in the guidance computers for essential flight changes, and to re-establish communications as soon as possible.
More details are provided below.
Let's start with the picture in the question:
058:04:03 Lovell: Houston. Okay. I want you to doublecheck my arithmetic to make sure we got a good coarse align. The roll CAL angle was minus 2 degrees. The Command Module angles were 355.57, 167.78, 351.87. [Pause.]
[This stage in the LM activation process was dramatised in the Apollo 13 movie. Fred has the three gimbal angles from the CSM's IMU which is relevant to the CSM spacecraft coordinate system. He has to do some arithmetic on them to adapt them for the LM spacecraft coordinate system. Those sums include taking account of the 2-degree docking index angle. Given the stress of the situation the crew are in, it is wise of Fred to ask the many minds in Mission Control to check what is otherwise simple arithmetic.]
The calculation is a rotation of one coordinate system (the CSM) to another (the LEM). This is a calculation that an undergraduate student might do in a linear algebra class. It is not a burn calculation, and certainly not a rendezvous.
Normal operations relied heavily on support from mission control, including computer calculations and simulator testing. Critical calculations were done in the Real Time Computing Center, the needed parameters radioed to the crew, and the parameters typed into the spacecraft's guidance computer. A complete lunar mission required about 10,500 computer keystrokes.
As Organic Marble pointed out, these parameters are written down (and usually read back) by the astronauts before being typed into the computer. The astronauts are given forms (PADs) to write down the parameters.
Some calculations were so difficult that even the guidance computers aboard the spacecraft could not handle them; they were left for the computers back on Earth. For example, the LEM ascending from the moon, to rendzvous with the CSM in orbit:
Calculating the velocity needed for this maneuver is more than simply raising the pericynthion, as other orbital parameters, plus maintaining the delicate relationship between the CSM and LM need to be taken into account. These complexities are beyond the capacity of the limited memory of the LM guidance computer. Rather than selecting a specific program to perform the Boost maneuver, the calculations will be performed in the Real Time Computing Center (RTCC) and relayed to the LM crew.
https://history.nasa.gov/afj/loressay.html (near the bottom)
If the on-ship computers can't handle this calculation, there's little chance that the astronauts would be able to do it.
The two essential functions, orbit determination and targeting, cannot be performed on board the spacecraft. The reason I am making such a big point about this is probably obvious. Putting together mission techniques with a G&N system like that is much more complicated that if the whole job could be done on board the spacecraft without external assistance. A tremendous amount of data must be relayed back and forth between the spacecraft and the ground, and the content and format of these data have to be complete and precisely compatible. Also, instead of only the three crewmembers being involved in the operation-that is, understanding and carrying it out- we must involve the entire flight-control complex. This makes the inflight job, of course, more complicated-but, believe me, it makes the planning job something else, too. Many diverse opinions about the planning task are expressed without hesitation or inhibition.
Mission control radioed parameters in advance of each burn. In some cases, they would send the parameters of two successive burns in advance. This was especially important for lunar orbital insertion, which happened on the far side of the moon, out of radio contact.
Apollo strictly followed flight plans which included "go / no-go" checklists. An example can be seen in the tables here. The loss of the guidance computer or communication was a "no-go" for all but the most essential operations.
Astronauts practiced only on those contigencies which had a contingency plan. Practice helped to refine these plans, but the astronauts never improvised a whole new plan. There were only 20 contingency plans:
By the time of the Apollo 17 mission, five distinct alternate mission plans, 20 contingency plans, and eight lunar orbit alternate plans were developed.
Apollo Program Summary Report, page 6-23
About 40% of Apollo astronaut training time was running through these 20 contingencies in the simulators. However, they did not have the time to test combinations of more than one emergency. This quote by Chris Craft illustrates that most situations were expected to be handled back at mission control:
In Apollo, as any other complex space mission, it is virtually impossible to develop premission plans for every contingency that could arise during flight. Although specific plans are developed for all abort potentialities that involve crew safety. most alternate missions are developed on a class basis by using the alternate test and mission objectives. However, the real-time mission planner is given a powerful assortment of mission-planning computer programs that enhance his ability to manage any contingency. By the proper use of these on-line computer programs, alternate mission plans can be developed in real time and can thereby augment the premission planning activities.
The contingency plan for a loss of communications was to continue using guidance computer (or the abort computer) programs, using parameters that had previously radioed from mission control:
The "sextant" isn't what you think it was. There never was a hand sextant. There were two devices in the CSM (the sextant and the optical telescope) and one device in the LEM (the Alignment Optical Telescope); all were bolted to the hull of the craft. All three were electronically read by the guidance computers, to calibrate the computer's representation of spacecraft orientation.
In the CSM, an astronaut moves the joystick, which (at a speed controlled by a switch) turns the shaft and trunion motors. The shaft and trunion are each connected to prisms in the sextant and optical telescope. This allows the astronaut to line up the instrument with a star. When the star is lined up, the astronaut presses a switch. The rotation of the shaft and trunion produces an analog signal (a potentiometer?) which is analog-to-digital converted by the guidance computer.
There's no dials for a human to read the shaft and trunion rotations. The devices are entirely dependent on the guidance computer. If the guidance computer is dead, you rely on instructions from mission control.
The LEM's telescope is first turned to one of six coarse detent positions. You then steer the ship (with the guidance computer!) until the star is lined up. You don't read a number on a scale; you press buttons to tell the guidance computer that the star is lined up.
The astronaut selects a detent and the particular star he wishes to use. He then maneuvers the LM so that the selected star falls within the telescope field of view. The specific detent position and a code associated with the selected star are entered into the guidance computer by the astronaut using the DSKY. The LM is then maneuvered so that the star image crosses the reticle crosshairs. When the star image is coincident with the Y-line, the astronaut presses the mark Y pushbutton; when it is coincident with the X-line, he presses the mark X pushbutton. The astronaut may do this in either order and, if desired, he may erase the latest mark by pressing the reject pushbutton. When a mark pushbutton is pressed, a discrete is sent to the guidance computer. The guidance computer then records the time of mark and the inertial measurement unit gimbal angles at the instant of the mark.
The crosshairs in the LEM's reticle can be rotated, and there is a dial that shows this angle. But this also involves manuevering the spacecraft:
076:24:39 Lovell: Hey, I bet I know how I could get an alignment. Give them a cursor spiral angle. Instead of maneuvering the spacecraft, I'll give them cursor spiral. [Pause.]
Through the eyepiece of the AOT, a crewman sees a graticule or reticle that shows a radial line and a spiral. The pattern can be rotated so that first, the radial line coincides with a star, yielding a 'shaft' angle, then it it rotated again to make the spiral coincide with the star which yields the 'reticle' angle. The computer can combine these to derive an accurate vector to the star. An alternative means of deriving a vector to the star is to manoeuvre the spacecraft so that the star crosses the X and Y lines, marking each time it does so. It is this method that Mission Control are not wanting the crew to use as it means taking the spacecraft out of its current attitude.
So, without a working guidance computer, you can't get the CSM's orientation, and the LEM's telescope works by changing the spacecraft's attitude.
Some maneuvers had more than one program that could be used. The contingency plan was to use another program. Of course, this depends on a working computer.
The CSM and LEM each had their own guidance computer and optical instruments, and if they didn't work on one spacecraft, the contingency plan was to use the set on the other spacecraft to assist. The instruments, the docking radar on the LEM, and the VHF ranging system on the CSM all could help:
Onboard the CSM, the Command Module Pilot was also taking distance and relative position marks of his own. Distance information to the LM was obtained from a VHF ranging system, where the LM broadcast a signal in the VHF frequency band that was received by the CSM's transponder. Angular data was obtained by locating the LM by its bright strobe light on the face of the vehicle, through the Command Module's sextant. With this distance and position of the LM, the CSM's computer could calculate the same position and burn information as the LM's computer could do with rendezvous radar data.
The guidance computers on both craft had the same design, and most software was shared on both craft. In particular, either craft could be used as the "active" craft of a rendezvous. A specific contingency for the loss of the lunar module's guidance computer was to get it into orbit (through the abort computer or manually) as best as could be done, and then have the CSM (with its guidance computer and support from mission control) come get it.
Again, this depends on a computer (although in the other ship).
Tracking was also performed from Earth, although much less accurate than what could be acheived on-board. If guidance is out on both spacecraft, this was intended as a backup. However, this depends on communication working with Earth.
Main and RCS engines were normally controlled by the guidance computer, including "manual" attitude adjustments. Although the astronauts controlled such manual maneuvers with a joystick, it still went through the guidance computer, which could throttle the RCS system using pulse-width modulation.
If the guidance computer failed, the CSM had a backup system called the Stabilization and Control System, which had its own set of gyros that could restore attitude (but that's not a main engine burn).
Failing that, all engines could be manually fired. The joystick went through the Attitude Translational Control Assembly to the RCS. In this worst case, they would need help from mission control to estimate the main engine firing times, requiring communication with and computers on Earth.
As you can see above, the contingency plan for a loss of the guidance computer was to rely on communication with mission control, and the contingency plan for a loss of communications was to rely on the guidance computer. They didn't train for both happening together, and frankly it would mean something even more severe like a complete loss of all electrical power or meteor damage. No wonder why the movie Apollo 13 shows them freaking out over powering down the CSM before the LEM is powered up.
You simply can't avoid using a computer, even if it means one on Earth.