I know exactly how you feel, from recent memory. I found that as I just kept at it, my hand gradually stabilized and my thumb got lighter.
But the way you've itemized it, that's a lot of multiple interrelated problems to be dealing with at once. You have to break it down into manageable chunks. And your exercise seems needlessly challenging to me.
Changing bass strings and moving around the fretboard is kind of a given here.
IMO, that's NOT a given! No matter how advanced you are, I think you're making this way too hard for yourself. You're learning a new technique here. And even more challenging--you're trying to unlearn programmed reflexes. Baby steps are what's called for. Really slow baby steps.
Don't even bother with damping at first--for example, just play a simple open string exercise like in Kitarologus, but make a point of giving your thumb a "home" string to touch after every activity. Don't worry about anything but finding the right string every time--not muting, not sound, not tone--just the touch; start as slowly as necessary, then gradually build to moderate quickness.
If that's too challenging (nothing wrong with that, and no one need ever know; but you have to be honest with yourself), just keep your ima fingers planted and only focus on the thumb, simple steady rhythm. Once that's comfortable, activate only one other finger, on a single string, first on a simultaneous beat, then once that's fluid, on alternate beats, then both. Then a different--single--finger in addition to the thumb. And so on. Just an example. Feel free to devise your own exercise. But the point is to isolate these new movements while your thumb and fingers learn them (and unlearn what they need to unlearn). (And this is all way before getting the left hand involved, btw!)