Most people take the balance weight off and leave it off.

If the slop were due to low or no oil in differential, the rear axle would be VERY noisy.

The most likely place for the slop are the flex disks and the differential mounts. The front flex disk fails first because it deals with much more heat. If you replace it, pay close attention to which way you put it in. It is marked in German, so look closely at the one you are taking out and compare to the new one.

Also shake the driveshaft up and down to check the center bearing carrier.

Good luck,