Hi guys, this is not a F series but it's a Ford.
I have a 02 e150 convertion van with 90k miles, I have been noticing that recently when I go to take a turn on dry pavement, when I applying power, like turning into my driveway or get into another street, I can feel a rumbling noise and vibration in the rear end of the van (it feels like if I were slowly spinning the wheels on pavement). Everything started happening after I went to Ohio to pick up a 24 travel trailer that weighs about 4000 pounds (about 1500 miles round trip), I've never felt it before when towing the trailer for the first time. It gets worst when towing but when I'm going straight I don't feel anything, it only happens in turns, loaded and unloaded. This is not a 4wd vehicle. I check the differential oil level and is fine, the color is amber. I tried to check the u joints but I have no idea on what to look there.
What could be the problem?

 

Change the rear end oil. Make sure to use the friction modifier or use amsoil gear lube. See if that clears it up...

 

I replied to your other post, check it out.

 

My E-150 did that last summer. When it got to the end of my driveway it went "CRUNCH, CRUNCH," as I was turning to get on the street. I backed up to the house and shut her down. When I got the cover off, I found several gear teeth in the housing. Mine has the 8.8 ten bolt posi trac. The side gears had given up so I pulled the third member and replaced all four gears, the frictions, steels, and bearings. That did the trick. If you get lucky like me, you might not need the ring and pinion. Those are complicated to get set up right the first time if you have never done one before. Anyway, the side gears in the third member are what makes the low rumble. Specially when your in a turn.