I have a tremendous amount of slack in my 1982 300D differential, and a fair amount of whine. The whine is not as loud now with Mobil 1 synthetic gear oil in the case.
While having the driveshaft down to replace flex disks, I made an effort to look at and measure the differential slack. With the rear wheels firmly in place (and thus the axle shafts not moving), the driveshaft slack is 60 degrees of movement (360 degrees being one complete revolution).
Now THAT is what I call slack!
Although a lot easier to estimate the measurement with the driveshaft off, you should be able to get close on measuring how much differential slack you have by engaging the parking brake and putting the transmission in neutral. Then, get under there and turn the driveshaft by hand back and forth.
Do I have the record for the most differential slack?