I would say that the flexof the tire in and out is a bit much, but that might just be the tire flexing under your gorilla strength.
Sounds more to me like a bushing or maybe a loose mount, like to a shock or leaf, or rear sway bar link.
As many have said, the "in and out" of the axle is normal, c clip movement.
I suggest jacking up the rear, removing the tires and rear rotor, this will give you access to the axle. There should be little no movement up down toward the front, or back, only a "sliding" in and out of the axle housing.
You might try oiling some of the bushings in the rear with PB blaster. Oil one bushing, then shake and test, and repeat. It would be cheap, easy, and if you do not find the issue, you now have some nice wet bushings.
Even spray the leaf springs
If you get under the truck and have someone else shake, you might be able to see, or even feel and give a better proximity to the issue.
My guess is a bushing.