I took off the back panel and found that one of the shock absorbers was broken. I ordered replacement shock absorbers and replaced them, with the help of the very useful videos on this web site. While I was back there, I replaced the (18-year-old, 1,500+ loads) belt. (I replaced the door strike too, since it was broken but not affecting this problem.) Definitely use a 1/2-inch socket from your socket set for the plastic pins, and maybe get a helper for the belt replacement. Watch the videos on this site!
It then spun but did not drain. I watched another helpful video here on checking and replacing the drain pump. I disconnected the drain pump and found a house key stuck in the pump. I removed the key and reconnected everything, and now my 18-year-old washer runs great again! Thanks, AppliancePartsPros!