Rear end problems
- rear wheel bearings
- drive shaft u-joints
- half-shaft (axle) u-joints
- differential wear/misadjustment
- worn out improper gear oil
With the wheels off the ground, I would start by feeling how much top in/bottom out movement there is for each rear tire; grasp the tire at the top and bottom and try to wiggle it top to bottom. This will give an idea for bearing wear in the trailing arms. You shouldn't feel much movement if any. While you're there, check the shocks and strut rod mounting points too.
Then with the wheels removed, try to pull/push on the trailing arms inward/outward from the differential. This will be an indicator of u-joints and excessive end-play inside the differential. Again, there should be minimal to none. (If you notice movement in the forward attachment of the arm, that could indicate worn out bushings.)
Grasp the each half-shaft and rotate back and forth to also feel for bad u-joints. There shouldn't be any feeling of movement that does not also rotate the wheel hub. Do the same for the driveshaft.
That pretty much examines all of the components that cause the car to move except for what's inside the diff. If nothing else has shown itself, before tearing apart the differential, I'd change the oil and put in real AC Delco rear diff oil and 2 containers of AC Delco Positraction additive. You can get this online but I prefer my local Chevy dealer. The clutches may be slipping and in need of refreshed lube.
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