The SuperHeavy/Starship does yaw before clearing the tower in a manner analogous to the Saturn V.
In the picture below (shamelessly stolen from here), the early tilt of the Saturn V away from the tower (in the y-direction) is a yaw maneuver to avoid the tower. It's designed to nudge the vehicle away from any arms on the tower that have failed to retract.
The Saturn V, like most rockets, then heads in an Easterly direction after takeoff (the z-axis in the next picture, from the same site), which means the vehicle also has to pitch in the z-direction soon after launch.
This may account for the "Texas-Two Step". The tilt of the vehicle in the z-direction then starts to increase with altitude in the direction of its intended orbital inclination. There are also roll maneuvers about the x-axis.
The SpaceX launch platform is to the East of the launch tower, not South as is the case with the Saturn V. It's clear from this video of the Starship launch on its 3rd integrated test flight that the vehicle begins to yaw only a few metres above the platform - see the graphical images at the base of the footage. At a tilt of a few degrees, the yaw slows, or perhaps stops, until it resumes with the first kilometre.
This diagram shows a similar tilt in the second Starship test flight in November 2023:
(NB: I can't find SpaceX documentation or an Elon explanation, but it's hard to interpret the yaw in any other way.)