I towed a 7200 lb 31' TT with a 2010 F150 Screw, 5.4 3.73 and 1857 CCC for 2.5 years. Never had a serious sway issue and I tow in the PNW and down into CA. I used a 4pt Equalizer WDH at 1st but hated the mess and noise. TT did have a tendency to wander a bit when passing semi's so I found a used Hensley arrow. Towing was as good as it gets with the HA.
Ended up trading up to a Ram 2500 CTD for several reasons. One was that the F150 with a RAWR of 4050 was at 4000 on scales. I had a canopy on it and the bed was loaded to the gills with gear. Also the 6sp trans was pure garbage. No way to lock out 6th or 5th so it hunted all day long. Plus the 5.4 towing 7200 lbs up in the mtns was a big dog especially at 4,000' EL and higher.
To bring all this forward to today if I had an F150 with the 3.5 Eco with Max Tow (Not the HDPP) I would have no issues looking at a TT in the 7000-7500 lb loaded to camp range. I would by a Pro Pride and get on with it. Much less than paying the $$$$ for a 3/4 ton along with the extra $$$ for gas as a DD. The Max Tow package gives you the 36 gal gas tank which would be perfect.