35' and 7500lbs+ is tooo much for an F150. The RK accentuates the problem since most heavy stuff is stored there. Getting a TW near 12-15% will be difficult.
A Hensley is not a solution if the TT is already swaying. It should be added to prevent possible swaying. If the TT is swaying it's not loaded right. If you're running C rated ST tires then you're at their max or possibly over. Are they 205 or 225? IMO you need to upgrade the TT to D rated tires and run 65PSI in them. Get the TW checked and make sure you have the front of the truck at the proper height. Read your owners manual for setting the truck up for towing.
Also if your Ford TSC is kicking in you have serious issues and should really think twice before towing again. I had a 2010 that had the TSC and never had it kick in while just driving down the road. Only time it kicked in was when I wanted to see how it worked and found a deserted road and purposely whipped the steering wheel left and right to try and activate it. When it finally did activate the TT was wagging pretty good. Not out of control by any means but none the less if yours is activating then you have major issues. I can't believe you haven't wrecked yet if the TSC is kicking in. I guess it's working.