A bit of play in the diff isnt necessarily an issue. Ive seen some that manage at least a half turn of one wheel before all the slack gets taken up and the other starts moving - check that it moves freely and try to have someone else turn the wheel wile you get close to the diff to listen for noise.
You mention a hanger bearing bush - recommend you take the driveshaft off if you can and check the bearing and all the U-Js. the section of driveshaft between the gearbox and the hanger bearing doesnt get much variation in angle, and a UJ running in a nearly straight line all the time can wear funny. Common problem on cars with independent rear suspension - youd think they would never wear out but running in a straight line is worse than at a modest angle. Could be causing your rumble / vibration at a specific speed. Check all the bolts and any rubber bushings too. Also a slightly worn wheel bearing can do the same.
Re oversteer - roll-oversteer could be making an appearance as your suspension wears - its when body roll causes a leaf sprung diff to cant left or right while going around a corner. Shocks and anti-roll bars may be loosing their effectiveness, coupled with tires + whatever else could add up to your situation.
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