Swaying issue

Usually a new set of drives will give the sensation of the truck swaying or the rear fishtailing because the tread blocks on the drives are really squirming around.

It usually takes a few hundred miles before the tires settle in.

The pulling may mean you need an all axle alignment.

I would check the front suspension drag link, tie rods and springs and hangers and also the rear suspension components too.

 

I agree with boneebone.

 

I'd be checking the suspension shocks, and springs check the wheel alignment as well. Unless your driving in really heavy winds it should not be swaying about.

 

New drive axle tires often promote sway, more so with a heavier lug pattern. Keep driving, the " cap" of your new tires will even out ( the center circumference) and the swaying will dissipate.
Next time you're at a heavy truck tire shop, look at the stacks of new tires and you'll see they all have extreme capping.
Running heavily loaded often makes the swaying nearly non existent because the tires " flatten " out on the road.