I went to Autozone like Og suggested (thanks Og!) and had a nice young man do a charge test on my poor cranky car. Turns out the battery is all topped, and his gadget didn't find anything wrong with the starter or the alternator. Which means (he did admit he was spitballing here) it could be a problem with the sparkplugs, or a sensor connected to the sparkplugs, or a sensor in some other part of the engine, or any one of a number of other things. (He didn't try to talk down to me, which was refreshing -- just gave it to me straight in layman's terms.) So I still don't know what's wrong, but I do know several things that most likely aren't wrong, and failure is still a result, so at least I have that.