Sounds like the differential is a gonner.
With the vehicle in neutral and the park brake set, check the pinion (where the driveshaft couples with the rear diff.) for any side to side and/or in and out slack. Slack will likely be accompanied by (much!)oil leaking from the pinion shaft seal.
If the pinion seems okay, I would suspect side bearings or C/V joints/shafts.
Rear diff. repairs are not complicated, but they are $$$$$, so good luck.
1980 240d 4sp
ASE Master Auto Technician
ASE Advanced Engine Performance