Judging by your symptoms my number one suspect would be the input bearing on the diff.
Seen this issue quite a few times.
Officer Douche Bag: " you know, I should give you a ticket for your exhaust being so loud"
Me: " In that case, shouldn't you be ticketing every Harley motorcycle you come across, surly they are louder then I am ? "
Officer Douche Bag: "Oh no, they came from the factory like that, it's obvious you modified your exhaust to make it louder, here is your ticket"
Me: "Have a great day officer"