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.
My big concern with my earlier post was for your safety if a universal let go, always worth a check if there is vibration.
Regarding the Idler bearing on the driveshaft, there shouldnt be any slop in the rubber bush, just a tiny amount and non in the bearing in its centre these are greaseable but the rubber bush does pack up. If worn it can cause vibration, but more likely vibration from the diff is damaging the idler.
Hopefully a mechanic will come on here and give better advice than mine
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