I'll bet the offcenter driveshaft is a result of the engine and tranny mounts being loose or non existant. The engine is likely skewed in the bay.

If you felt something shift it could have been the engine, and caused something to scrape or hit. Perhaps the "park" like feel is that something tight up against some frame componant.

I would certainly straighten out the engine, run the bolts through the engine and tranny mounts. It's one thing to be missing a few engine mounts but that combined with no tranny mount is asking for big trouble. Torque is causing the whole thing to squirm around in the bay!

Once you do that then I would take it out on the road again and run the tranny through the motions. You still likely have issues there as well.