Many have the attitude of, "It's a heavy duty truck, I don't need a weight distributing hitch or sway control."
You might not need it in the "by the book" sense, as you have plenty of axle capacity and receiver capacity and payload for what you are towing, but what a WD hitch with integrated sway control does is stiffen that ball joint between the truck and trailer, making it go down the road like one long unit, like a bus.
When the bow wave hits your trailer without the sway control, it pushes the trailer off line just a hair. Because the trailer has so much leverage on the truck, it yanks the rear end of the truck, then shoves it as the trailer comes back in line. That is the push-pull that you feel.
Sway control prevents the trailer from getting blown out of line in the first place, or at least minimizes it, reducing or eliminating push-pull entirely.