As I picked up the mug with my right hand, slowly, controlling the motion to reduce the tremor (and get more tea in my mouth and less on the floor), I watched my hand and arm perform an action that has been automatic for about 40 years. That was when I noticed that I wasn't doing something that should have been automatic: I wasn't articulating my wrist at all. Try it: raising a mug without moving your wrist is awkward. Here was a key to being a bit less of a spaz in public.
This offers me some more insight, or confirms what I thought, about the brain damage I have: as well as affecting some muscles directly, I've lost some of the learned components of complex movements, like picking things up, or walking. I suspect this is where the tremor originates. To simplify: one part of my brain has the intention to drink some tea, and devolves the task to other parts of the brain to make the movement happen. They, in turn, devolve further, and ultimately direct the muscles to move.
I've killed off some of those intermediate bits of brain that handle the automation of the complex muscle motion, so a higher layer of abstraction has to hustle to control the muscle directly, overcompensates and overcorrects in turn causing the tremor. If I concentrate on a deliberate quality of motion, I'm using other parts of the brain, engaging the musculature more actively, and have less of a tremor.