The axle shafts are supported and stabilized by 2 sets of bearings:
The outer ones (located in the hubs) - wheel bearings. And an inner set (inside the differential where the axle shaft mates with the ring gear) - the carrier bearings.
Since you feel that the outer bearing sets are tight, the next step is to check the carrier bearings.
Drain the gear oil from the differential and remove the cover. You'll see where both shafts enter from the left and right. You'll also see the bearing caps that hold the carrier bearings in place. Here they are with the caps removed:
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With the rear end up on stands and the differential cover off, start the engine. Put the gear shift in drive and crank the idle up a bit. Go back and look inside the differential.
See if you can notice the ends of the axle shafts moving in any direction (other than rotating - duh). They should be stationary. Same goes for the carrier/ring gear assembly it should be rotating smoothly. Have an assistant rev the engine a bit to test at higher speed.
Don't worry about lack of lubrication , you can safely do this procedure for extended periods - as long as the wheels are up
Take your time and watch closely. Shift the transmission into reverse, watch for free play and then shift back to drive.
Any runout, wobbling, up/down or back/front movement is an indication that the bearings are played.
If everything seems tight - IDK.
Make sure you inspect the inner face of the drive shaft flange where it makes up to the brake drum and also check that the gasket is flat and smooth:
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Both of these surfaces are designed to be machined flat and true. Any excessive rust/debris that gets between them will cause the drum/wheel/tire to run untrue.
Here is an exploded view of your axle:
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