Okay, phase three. I drove to town, and bought a multi-meter, and then drove home (25 miles). The turn signals were working, but slowly, and dimly. The tach was working, but the temperature gauge was not. When I got home, I checked the voltage at the battery posts, while the engine was running. The voltage was 9.6, or thereabouts. Ugghhhh

I turned the car off, and when I tried to start it again, the battery was dead. It appears that either the battery is gone, or the alternator isn't charging. I've got a charger connected to it right now.
What's next? It's a sealed battery, so I can't check the battery density.