Well, today I cleaned up the old sending unit a bit (not a lot), and got the float arm swinging freely. Oh, the float is about 1/4 full of gas. With the unit outside the tank, I hooked it up with jumpers to the gauge wire and a ground, set the float to the high position, and sure enough, my gauge read full. Mystery solved (mostly). I was considering just putting my new sending unit in when I get it and call it good.
Then I put a flashlight and my camera up to the hole and looked at the inside of the tank. It is hard to get a good view, but the sides and upper ridges on the bottom looked pretty good. But there appears to be gunk in the bottom of the lower ridges. There is also some sort of box looking thing hanging from the top, sort of in the middle of the tank. Maybe a baffle of some sort? This seemed to have black crud on the face of it that I could see.
If you recall from earlier in this thread, I mentioned that I had run out of gas, but was only able to put about 10 gallons in. So I'm thinking that maybe the sediment crud in the bottom of the tank clogged up the filter on the pick-up tube. When I put more gas in, it may have temporarily dislodged it and I was good again.
So I've decided to bite the bullet and get a new tank.