Sounds like you don't have enough tongue weight or your hitch isn't set up correctly. You need to go to the scales or find a reputable shop help you out. Also make sure your truck tires are properly inflated. Too little information given to speculate further. I know F150s which how towed units larger than yours satisfactorily in terms of stability. The new 5.0 should provide enough power.
Additional thought: If you have the Equalizer BRAND hitch you should be set for weight distribution and sway control. EQUALIZER is a well-regarded hitch. My personal experience with them is great.
As a simple thing to try:
Check height of front fender to ground before and after hitching the trailer. It should stay close to the same. It if is going up too much then not enough weight is beind transferred with the WDH. Adjust the bars if needed.
You could also try moving weight to the front of the trailer.
Good luck.