1) Tongue weight
2) Tongue weight....a trailer has zero idea what it is being towed by. It only knows it's connected by a ball. A proper setup won't have sway. If soft tires start sway, the trailer shouldn't continue it.
3) loading. 1000 lbs over the trailer axles is not the same as 500 lbs 10 ft fwd and 500lbs 10 ft back, even though the CG and tongue weight would be the same. Inertia, pendulum motion, etc. Center your load over the axles.
4) More tension on the WD bars creates more friction which dampens sway, but again it shouldn't sway if properly loaded.
That's plenty of truck for that trailer if it's only you and your wife. And J2807 is NOT only pulling power as someone said, it is quite a bit more.
Let me renumber:
1) loading
2) loading
3) loading
4) everything else (tires, hitch, tongue height, etc)