What you are experiencing is so common it has a name, “drive line snap”. I’ve been in and around Chrysler products for 45 years. Some don’t do it and some do it worse than others. Some do it so bad that just letting off the gas (or diesel) at 15 or 20 mph and hitting the pedal can cause an audible metallic clank.
I have rebuilt transfer cases and differentials and there is not much inside of a transfer case that can contribute much to the problem. There is plenty inside of differentials that can produce slop and noise and the u-joints can contribute too.
Look for metal particles in the differential oil when it is changed. Best to remove the differential cover so you can see what is in the bottom of the case. If the oil is fairly clean and there is not a quarter inch of sludge in the bottom of the case then the differential is just a little lose and probably okay. If the pinion bearings are going then the pinion seal should start leaking or at least weeping. Of course, the pinion seal can go bad with out the bearings going bad (unless it’s run out of oil).
If the u-joint(s) are going bad then you could hear a chirping, squeak, or rumble and or feel a constant vibration above, say, 40 mph. I would just drive it. If the u-joints are not making any running noise or vibrating and the pinion seal isn’t leaking you are probably okay. Parts may have just broken-in in the rear differential a little on the loose side and there is nothing to worry about.
I had a 1974 318 Duster that had 237k miles on it before I got rid of it – same color as Al Bundy’s Dodge Demon

– lots of drive line snap. You could put it in park, jack up one rear wheel and turn that wheel over 1/8 turn of play yet the oil was clean (no visible metal) and the pinion didn’t leak.
