I had kind of a similar problem several years ago. It was not as bad as you are describing but did so less severely. I took it to a powertrain company and they took the rear differential apart and could find nothing wrong with it. I continued to drive it and it continued to get slowly worse.
I eventually took it into a Ford dealership and they couldn't find the problem either but didn't give up. They put it on a dyno and just kept testing it to try to find where the problem was coming from. After several days the problem was found by a failure in the differential.
The carrier was cracked and was not easily visible when not under load. When the Ford dealership kept running it under load it finally broke all the way through. I ended up having to replace a lot more than the carrier because of all the damage.
I don't know if you have a similar problem but it is something to consider.
2002 FX4 Bright Red Supercab 4.0 SOHC, manual 5-spd & Tcase, 33" BFG MTs, K&N air filter, Modified air box inlet, Gibson SS cat. back exhaust, Bama flash, Hidden Hitch round receiver, Rear helper air springs, Electric trailer brake controller, Line-X bed liner, WAAG bar with KC lights and Wildernest shell.