Regardless of what everyone is saying, you simply cannot turn an ear of corn into a turnip. Adjust, get new tires, different suspension, even put on sonar if you want, it's still an F150. It will never be anything more.
You have 2 things going against you.
1. Truck too small and trailer too big. (tail wag the dog, that will never change).
2. No matter what you do to your truck, you have now developed a physiological and mental distrust for the truck-trailer combination. You will never overcome that distrust and have complete confidence in your equipment, even by doing your best to compensate altering the truck. Once the child has been bitten by the dog, he's always and forever Leary of dogs.
Solution? You more than likely will continue using your current rig combination and more than likely, 10 minutes before "take off" you'll be making a mad dash to the bathroom for a good sit-down where your nerves will do their finest on your physically. But, you'll continue with no changes, or the least evasive changes possible, until a year from now you will either quite towing the trailer completely, or break down and swap for a heftier truck.
Unfortunately, this is the way it is.
I wish you good luck! And do hope your nerves have more stamina than mine do!