I had a crapload of saves, not counting backed-up ones, so I erased maybe 600 extra things in the saves folder (out of 800) and the problems seem to have gone away. Seems like a lot of hissy-fitting over couple dozen extra saves. I won't say its definitely fixed yet (give me an hour or so of playing and saving to see if it returns), but it hasn't combusted yet.
This is quite interesting. I did a test a long time ago and found that Morrowind starts to suffer severe memory corruption if you have too many saves in your save directory, But it has to be a lot of saves, on the order of 800 or so, and that seems to match what you are reporting. I strongly recommend not keeping more than a couple hundred saves in the save directory at one time. Personally, I only keep about 100, at most.
Also, whenever you see this error: "Trying to load a bad form in TES3File::NextForm", it means the file Morrowind was trying to read at the time is corrupted OR possibly that Morrowind has gone braindead due to memory corruption.