So, I took it apart. It's a two faucet shower, unknown brand. The washer on the end was chewed up, and the seat had a nick in it.
I'd have replaced the whole assembly, but nobody in town had one. So I rep laced the seat and the washer, new packing, etc.
Turned the water back on, and it still dripped at the same rate. I took it apart again and swapped out the cone washer for a flat washer from a rebui ld kit, but it made no difference at all. It still drips. Aside from the drip everything works fine. The water turns on and (mostly) off smoothly, no leaks, etc.
I'm not sure what to try next. There isn't easy access to replace more par ts, but really the stem repairs should have done something.