The amount of play you mention is about right on a 3.89. Since you mentioned a leak, this could be as simple as the rear is out of gear lube.
Depending on what is leaking, you probably are going to have to pull it out.
There is a pinion seal and the gasket to the banjo housing is just a paper gasket.
While it is out, you can check for play in the bearings and check all the bolts for tightness. It will also let you look at the wear pattern on the ring and pinion which will indicate if the issue is wear on the ring and pinion where they engage.
If you have been running without lube or very low lube, any of these scenarios are possible.
While it is out, check the play in the axle bearings as well and make sure that the backing plate gasket and the axle seals are good. It wouldn't be a bad idea to replace those while it is apart either.