I remove the handle, open up the escutcheon plate, and find there's a turnoff for the hot and cold before the mixing valve. Turning those off stops the trickle. Good. Take it apart and find the replaceable cartridge. Drive to Community Plumbing Supply in Springfield (BTW, if you want something like this, screw the big box home improvement stores and find yourself a real plumbing store. Community Plumbing rocks, they have everything, lots of stuff Home Depot just doesn't have.), they have the replacement cartridges, I'm upset it's $70 but the alternative is ripping open the wall and replacing the entire assembly, which would cost a few hundred, and take our only shower out of commission for several days, so this is good.
Take it home, put it back in. Turn on just the cold to test, and nothing. Weird. So I turn on the hot too and then it starts working. Turn off the cold and it completely stops (and yes, I have it set so it's warm water and both are feeding.) I realize that the cartridge is a pressure balancing one designed to prevent scalds if, say, the cold water goes out or its pressure goes down due to a toilet flush or whatever. This is good. The other weird thing is the old one would go from cold to hot when the handle was turned, this one goes from hot to cold. Shoudn't take long to get used to.
It works great and moves much more smoothly. I also was able to really clean the parts with some CLR and elbow grease and now they're all shiny bright. Also got out the dirt you can't normally get to so it's much cleaner. I resealed the escutcheon (that's such a strange word for "the metal plate that covers a plumbing fixture's interface to the wall") with caulk and it looks great.
I feel so, well, handy.