2014 F150 E-Rated Tires for Towing

I pulled a similar trailer with a 600 lb. 4 wheeler in the bed of an F-150 (probably over-loaded) with 255/70R-16 tires for years. I just used friction sway control and never felt out of control. I doubt it is just tires, although stiffer tires might help a little. OP needs to weigh the rig hooked up and then the truck alone (truck loaded the same with out the trailer). The weight goal should be 10 – 15% of trailer weight on the hitch and no weight change on the truck front axle, hitched or not. If the front gets light when hitched, this could cause a sway feeling. WD bars should be adjusted for the same front bumper height to slightly up (less than ¼” up) on a pickup. OP--is this the first trailer? Could you just be feeling the bow wave of passing vehicles or the out of control feeling of sway? A bumper pull trailer will always require some steering correction. But, it should be one or two turns of the wheel—not a constant back and forth.