I have the same AP and chartplotter setup, just not a hydraulic drive, and have not experienced what your describe. However, I never let an absence of experience prevent me from offering advice
I assume the two types of course following you describe are (1) AP set on a heading by just hitting the AUTO button on the ST6000 control head
; and (2) AP set to a waypoint by activating a route
in the C120 and having the C120 control the AP via SeaTalk
It does sound the like heading used by the AP is changing with boat speed, but why that would happen is a mystery to me. I can see how a rudder position sensor problem might cause the AP to over and under compensate for heading changes, but not set the boat on a new heading.
This problem might happen if deviation on your boat is changing with boat speed, I just can't guess what might be causing that to happen. I don't believe it is a problem with the deviation table calculated by the AP Course Computer in calibration mode. I think this table is used (with variation) to report a true heading on the display, but AP maintains course with the uncompensated heading from the fluxgate.
So I can't offer any solution. My only recommendation is to carefully collect more data on the problem once you are re-launched this spring. Be sure the association with boat speed is real not just a coincidence.
Although I don't think the deviation table in the AP is the problem I would definitely re-calibrate once you have reproduced the problem. Check the location of your fluxgate compass - is it near any thing that moves with boat speed (e.g., free wheeling prop shaft) that could generated some minuscule magnetic field?
If all else fails reset the AP to the factory default (after recording all of the settings) and see if that changes the problem. If it does then one-by-one restore those settings until you find on that is associated with the problem.
After that it is time call in a real expert.