Most satellites do have a thruster system of some sort on board. Indeed, the lifespan of a satellite is often determined by how much fuel it can carry, because once it runs dry, it'll start to slip out of its orbit due to a number of physical processes (for example, drag from the few atmospheric particles that get up that high, sunlight pressure, and the earth's slightly asymmetric gravity field). Most of the time, when the fuel is running out, we spend the last bit to kick the satellite into a "graveyard" orbit (an area set aside for nonfunctional satellites where they'll stay well away from the active orbits and are unlikely to collide with anything), or drop the satellite's lowest orbital point far enough that atmospheric drag will eventually bring it down (which may take weeks or months).
There are a lot of thruster technologies, and only some of them resemble a rocket. Hypergolic propellants have historically been popular for thrusters; in that case it's a tiny rocket where the fuel is two materials that spontaneously ignite when they're combined in the combustion chamber, which means you don't need an igniter system. The earliest satellites used cold gas thrusters (basically using an aerosol spraycan to get around) and those still get used from time to time. The new hotness is ion thrusters, which don't really look like a rocket at all. They are incredibly weak and power-hungry, but extremely fuel-efficient, so if you're willing to spend days or weeks inching your way into the proper orbit, that's an attractive option. (A notable user of ion drives is SpaceX's Starlink constellation.)
What you're looking at, though, is a cubesat, which is usually sent up as a ride-share with some larger payload that doesn't completely consume the rocket's lift capacity (for example, if you're lofting an 8000 lb satellite and the rocket can comfortably lift 8400 lbs to that orbit, you can pretty easily add in a cubesat frame and sell those slots to other people). Cubesats are typically tiny -- each "unit" is a cube 10 cm on a side that would fit comfortably in your hand. Most cubesats are one unit, though some of them are two, three, or four units stuck together -- but still no bigger than a shoebox. In any case, cubesats often don't have any drive system other than the spring that pops them out of the rocket, and they're intended to reach only an approximate orbit and last only a short time. "Haphazard" is a fully appropriate word. They're mostly meant for university students to get some experience with satellite development or get an instrument into orbit on the cheap, so nobody's going to be overly upset if they're off their intended orbit by a few kilometers.
Some cubesats have been pulling off some really neat tricks with orbital maneuvering, though. At that tiny size, it's possible to angle solar cells to use as a light-sail or reposition in orbit by effectively pushing off the earth's magnetic field.