Towing issues specific to loaded Toy Haulers

We used to have almost that exact set up... F150 with a bumper pull hauler with very similar specs to what you mention. We tried all kinds of stuff to make it a comfortable experience... WDH, sway bars, air bags, etc etc. In the end we traded up to an F250 (it was easier to do that than try to ditch the trailer). It worked so much better once we traded in the 1 year old F150 for an F250 (yes we now have another F150 but just as a daily driver), we didn't feel like we were getting pushed around all the time. I know you said you're not getting another truck, but just wanted to share what our solution was to the same problem. As you can see in my signature, we've since upgraded to a 1 ton dually and a 5th wheel.

Toyhaulers are designed to have weight behind the axle.... so that is likely not your problem.