Geez, that bit about the diff going up and down. The problem has to be somewhere between the back end of tranny and the differential. That doesn't leave too many places to check. (Well, maybe problem in diff or axle but I'd look there last.)

Get the car back up so you or someone can look around while it idles in gear.

Tranny tail housing has a bushing that wears. It's possible the bushing froze to the output shaft??

Plus any and all flexible/movable parts between there and the diff could freeze or shaft could be bent. I'm sure that none of the above happens very often but obviously one of these components has gone bad. An inspection should find it.

Might have to leave tranny in neutral and spin driveshaft manually, slowly to see what it's doing.

If a shop did the looking when it was on the lift and did not find the problem, I think that would mean the shop does NOT want to work on your car. Maybe they don't like Volvos. Too bad for them.
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DAMHIK: Don't Ask Me How I Know - - - Sven: '89 245, IPD sways, electric rad. fan conversion, e-codes, 28+ mpg - auto tranny. 500 mi/week commute. '89 245 #2 (wifemobile). '90 244 (spare, runs).