Here's a follow-up to this question. Firstly, the brake light was not on when I started the car with the park brake off. It stayed off while I backed out of the driveway, but always came on once I put it in drive and started moving. It did the same thing every time.
I changed the rear drums and shoes because there was some minor squeeling on the rear right drum. There was lots of shoe left on both, but the right drum had some deep grooves.
So, after fighting with springs and brake dust for a few hours, I had new drums and shoes in the back. Funny thing is, the brake light is off now. It works when it should to tell you the parking brake is on and then goes off when you release it.
I don't get it. There are no sensors in the drums, and the old shoes were not that bad, so I don't see any way for any sensors to detect bad brakes. The only thing I can see is that while doing the brakes I was tugging on the park brake cable and either freed something up or reset something.
Anyway, I am happy to see the brake light go off when I release the parking brake.