First, get the truck driving without problem without the trailer. Might be front end problem or might even be simple as insuring that all tires are full of air. An F350 should have no problem handling that trailer.
Second, you do need to have tension on the bars to balance hitch weight on the truck suspension. You should not be able to hand lift the tension bars from the hitch.
Third, the weight in the trailer should be balanced also. If all the weight is in the back of trailer then it will tend to sway with you.
Sometimes trying too hard to control the trailer causes more "activity" than it should. The more you turn the wheel, the more you will need to turn the wheel.
Good luck.
