You'll need a new gasket and some good old 80-90 gear oil.
Chock the front wheels. Chock them again. Jack both rear wheels off the ground. Pull the cover off the rear diff. Check the oil for pieces of metal, and look at how glittery it is. Glitter is very, very bad. The Hoe is a man truck, and abhors anything not manly, especially glitter. You should have a magnet on your cover, check it for more glitter, and/or pieces of gear teeth. Again, glitter is bad.
Spin one wheel until the carrier is open, and you can see the spider gears in the middle. They're the first thing to go. If you can see damage, you need to get to a rear end shop really, really soon. Bend over, because the rear end shop is going to put it in your rear end in a very glittery sort of way. Glitter is bad.
If your rear end has no glitter, and supposing your Hoe is a 4x4, you might also check for play in your transfer case. But since you say that with the drive shaft locked, you have a good amount of rotation in your wheels, I'd go for the rear diff first. This rear end was never meant for towing, or doing anything cool. They made an effort, but fell way short. If you're towing, you need to gut your diff, and go with a detroit product. Eaton made the glittery gov-lock, and for that I hold them eternally responsible. Stupid glitter.