Sorry for the delayed answer, I'm in Europe and connectivity challenged.
If you have a way to raise the rear wheels safely, move the driveshaft by hand and see if the rubber flex disks are not tight. The diesels don't make enough torque to hurt them, but these cars are old enough now that I would think that one of them could be rotten and has fallen apart.
'84 Euro 240D, 523K miles
'88 300E 5 Speed
'81 300D Daughter's Car
Over 800,000 miles in