The difference between a F-350, F-250 and F-150 if you are under 10,000 GTW is not all that much. 38' length X 8' tall or 10' tall if the trailer is that high of sidewalls is into almost 400 square feet of sail area, a LOT to deal with.
Having a F-350 with a slightly higher rear axle weight might help a 'little' but not nearly as much as you might like. IT is the front of the trailer being twisted left and right by all that wind of a passing truck, and that 400 square feet of sail area doing the pushing.
Your truck rear axle weight might be in the 2,500 pound empty range, while a F-350 gas truck might only be say 400 pounds heavier, so both will tend to get pushed around a bit on two lane roads by the 18 wheelers going 70 in the other direction!
You might look into the Pro Pride hitch system. It was developed by a engineer that had a horrible experience towing a travel trailer behind his then nearly new 1950's sedan and 1950's travel trailer. Many years later he designed a much better hitch system.
Good luck,
Fred.