If your real lucky, maybe a bad u-joint. Undo the rear u-joint, and rotate by hand. Is it dry, or is the joint oily. Also check the front u-joint. While you have the drive shaft off rotate the pinon gear, try to shake up and down.
If your lucky, maybe the pinion gear flange nut has come loose, and just needs retightened. If your differential is the 489 case it has a crush sleeve (may have a large "9" on the drivers side of the case). If it is the 742 case it has shims, and may have a large "2" on the case.
For the crush sleeve, we used to change the pinion seal, hold the drive shaft yoke with a large pipe wrench, then tighten the pinion nut to take up all the slack, then a little more...like rotating from 1 oclock to 2 oclock. Put loctight on the CLEANED pinion and nut threads before tightening.
If its the 742, hold the drive shaft yoke, and tighten with torque wrench to specified torque.
If this does not work, sounds like you have something coming apart. I would then look for rebuilt differential assy, or find one in the parts yard. We have lots of them in the parts yard here, so may find one at low cost.
After you remove the old diff, then clean the inside of the axle housing real good, all the way down the two axle tubes to get out any debris. After you go back together then drive it maybe one hundred miles and change the fluid again.
We need more test data to help isolate the trouble. I would NOT climb under the car, even on jack stands, and run the engine/drive train, although this could help isolate the problem area.
Hope this is of some help.