Progress!
I found that the non-drive side had an anti-spin washer and that it was installed backwards. That's how it came to me, I don't know how the previous owner managed to get it put together.
In any event, everything fits together MUCH better. The axle nut on the non-drive side is suspect, I think that it has become too worn to hold under load. Even threaded all the way on, it actually popped off at one point as I worked on the other side. Hopefully a new nut with nice new threads is the end of this episode.
Thank you to everyone who posted in response, it was a huge help getting this done.