I'm mostly posting this here because I was unable to find much useful info about this across the internet, discovered this quick and fairly easy fix pretty much by accident, and this seems as good a place as any to document it.
It seems to be a common problem with older AN1x synths that the ribbon controller starts sending erroneous CCs. In the worst of cases, the ribbon ends up behaving as if there were a finger stuck on its very leftmost side. I can only guess that this comes from a deviation in resistance caused by wear within the resistive carbon sandwich that makes up the ribbon, which in turn messes up whatever is supposed to determine whether or not there is a finger present. Most other posts suggest swapping out the ribbon and corresponding control board, but in my opinion this is not necessary, and in some cases doesn't even fix the problem.
I have not taken to the ribbon control circuit with a multimeter or analyzed it to determine exactly which part does what. Therefore I don't know exactly why this fix works, but it did the trick for me.
Open up the synth by unscrewing ALL of the black screws on the bottom of the device. They are all identical, so don't worry about what goes where. When it's open, you will see this:
(image not mine)
What worked for me was turning the outermost trimpot on the ribbon board EVER SO SLIGHTLY (a nearly imperceptible amount) counter-clockwise. I had the synth resting with its rear panel on a chair, so I could simultaneously mess with the ribbon and adjust the trimpots. I just turned until I reached a point where the ribbon weirdness went away, and then a tiny little bit more for good measure.
Hopefully this will help someone with the same issue.