My truck has this weird vibration coming from the rear when I am at a full stop and slow accelerate.It is especially noticeable on cold days and when the vehicle has not warmed up even though it can still happen after warming up.

I have taken it to Ford to see what the issue was but they wanted me to replace the brakes and rotors first. Even though the brakes and rotors only have 40k miles on them. I didn't understand why they couldn't put known good parts on them to see if the brakes are the reason. But if I went ahead with the replacement THEN they would send it over to the Driveline specialist to do further diagnostics.

I have replaced the transmission oil, seal, and filter around 120k miles. Stupidly I didn't change the fluid at 60k. I replaced the Rear Diff Oil around 80 or 90k. 

Specs: 2011, F150, Extended Cab, 6.5ft bed, 140k miles, v8, Rear Wheel Drive.

Thank your help.

If I have left any details out I will answer them as soon as I can.