I switched out the OEM Wranglers for E rated Michelins on a 2010 F150 XLT 4x4 5.5 bed. Don't expect your truck to feel like it's on rails with the E's. Yes they do help but you still have a 145" WB pulling a 7000+lbs 32' TT. I would look at the TT 1st for tongue weight. Also you don't say what type of sway control. Are you using those cheapo friction pad add-ons or a built in WD? I towed almost the same exact size TT (7200lbs @31') and even with 900lbs on the tongue, E rated tires and a properly setup EQ 4pt sway WD it still was not near as solid as with my 12 Ram 2500. IMO it's just the nature of towing when you have as much TT as you do truck. When you get more truck than TT that's when things like sway become less apparent. I could tow our last TT with the RAM and just a simple round bar WD w/o sway control as well as the F150 dialed in.
With all that being said, I will say going to either D or E rated tires help. The factory GY's are just to soft. They are there for a reason. So 1/2 owners get that cushy ride. That cushy ride is exactly what you don't want when towing.
