I had similar issues with my f150. Mine was not an HD and I was near my weight limits. I never got it sorted out and after a scary drive upgraded to a 3/4 ton. I haven't had a problem since. I see and talk with so many people towing with f150s that have no problems, but from time to time talk with people who have had the same issue you and I experienced. Unfortunately non of them figured it out. Hopefully you will be the one who finds the answer. My best uneducated guess is that it has something to do with the truck's suspension. Good luck