Somethings definately amiss if you're having any sway or push-pull using a PP with your F150 HD. That truck should have no problem towing that TT.
Is the truck moving or just the TT waggling some?
I towed a 31' 7200lb TT with a Max Tow 145" WB F150 using a Hensley Arrow and had zero push pull.
Only thing I can imagine is that with your TT you have too light of a tongue weight. It comes from the factory with a marginal tongue weight, then most all the heavy stuff is back of the axles.
Have you ever weighed the tongue?