This will be hard to kind of do with the center section still in the axle/accurately. If you block the rear tires on one side front and rear on a flat surface, with the rest of the driveline in neutral, there should be hardly any movement (rotation) from the pinion output since the carrier(ring side) should be locked. If that makes sense. You should be able to detect a small amount of backlash spacing between pinion and ring gear. Have you tried drawing the diff fluid to see if anything extra come out?
Another thing you might want to consider is the possibility of a small bend in one of your wheels, not sure what you have access too as far as tools, but I have spent more time than I care discussing tracking down similar issues. A slightly out of round wheel can "balance" on a machine, but upon closer observations depending on the tech doing it or if you have access to a machine, may in fact reveal that the wheel is out of round- not always noticeable or easily diagnosed.
I feel your pain needless to say. Good luck.