depressing newbe question of the day

Your video of that u-joint moving radially tells the whole story. It's old and got beat up by the lack of a bolt. On final assembly, clean everything and use medium strength loctite.

U-joints are cheap.
Just call around locally and see how much someone will charge you to change them out. You can carry the driveshaft to them. It's fairly simple and maybe you can watch them and learn.

Change both. If you chock the front tires and jack it real high in the back the tranny fluid should stay put.