Update: did a thorough inspection of the drive line. The transmission mount has completely disintegrated, causing the tranny to rattle against the cross member. There is no noticeable vertical play in the differential. The rubber part of the trunk mount is, however, resting on the washer. I'll replace the transmission mount as it's the most obvious problem and see how much slop and noise goes away. The flex disc and center carrier are OK.

I will also check the transmission output flange nut for looseness. Is a special socket necessary to tighten in?