The bushing that centers the prop shaft where it bolts up behind the gearbox on your rig has packed it up. There is a “nose” this bushing inserts into. The symptom you describe is a classic symptom of this bushing requiring replacement. The part sells for $10 bux on ECS.
Next, because of this untimely centering bushing failure, the center support bearing has likely been exercised to the point of being damaged. The thin rubber “webbing” bridging the inner hoop to the outer bracket, it does not take much to damage it. Unfortunately, this is $133 bux. Get it, do not skimp on this. Lastly, when you remove the two-piece driveshaft, the location where the male splined end is in the female part, on the outer portion of the driveshaft, YOU MUST use soapstone or say a white-out pen to mark its original join-up location, because the driveshaft is balanced. When putting the driveshaft back together, because the center support bearing install requires this, you MUST reinstall the halves with the mating mark exactly how it was when you disassembled it. I’m not going to kid you, doing this job on your back sliding back and forth on cardboard is going to blow, and if you get weird about having a 3500 pound car inches away from your grill like me, you’ll quickly catch my drift. Good luck, hope his helps.
BMW Part #26117526611
Link:
https://www.ecstuning.com/b-genuine-...CABEgJnovD_BwE
Link:
https://www.ecstuning.com/b-genuine-...CABEgJTAvD_BwE