My F350 pulls multiple trailers ranging from 5000 lbs tag to 25,000 lbs gooseneck. The truck handles all but 1 trailer very well. The one that behaves like yours is my 5th wheel when I have my motorcycle carried on the back. Take the motorcycle off and it pulls beautifully. With the bike on back and no counter weights up in the bedroom the trailer is very dangerous to tow even though it still has 2000 lbs of pin weight. By adding 400 lbs of ballast up in the bedroom the trailer pulls ok although not the way it does without the bike on back.
It's possible a heavier truck won't accomplish what you want. I would borrow a 1 ton on a windy day and see if it solves your problem before pointing my finger at the f150 as the entire problem. Maybe the solution is to move the trailer axles back 2 feet.