Pictures would help a lot !!
Somewhat accurate weights would help. We are just guessing without them.
Sounds like you have more than one issue. Could be a trailer loading issue, could be truck over loading, could be undersized hitch, could be impropper hitch set up.
You need to be aware of more than just a max tow weight rating. There are also ratings for what the truck can carry (payload) and a weight rating on the hitch receiver. That max tow rating was set, considering the driver only weighed 150 lbs, and he had no aftermarket accessories, no passengers, and no cargo.
GVWR (gross vehicle weight rating) is the maximum weight rating on the truck. I would suspect, you're over very close and possibly over this.
Payload is the difference between what the truck actually weighs, and its GVWR. Payload is eaten up by everything and everybody in or on the truck. It includes people, fuel, cargo, aftermarket accessories, weight distribution hitch, and trailer tongue weight. There should be a sticker on your drivers door post, that shows you what the payload was, when the truck left the factory. Any aftermarket changes, would affect that number and the max tow rating, pound for pound.
Receiver rating is max weight on the hitch, with or without weight distribution.
You say the truck has 6500 tow capacity (propper term). If you put 6 - 700 lbs of people and cargo in that truck, it's tow capacity is less than 6000 lbs.
On the trailer:
It could actually weigh over 6000 lbs.
You guess it weighs 5700 lbs. That is estimating only 450 lbs for a camping load. The average camping load (dishes, pots and pans, bedding, clothes, camp chairs, BBQ, groceries, water, etc) is 800 to 1000 lbs.
On the hitch:
The "V" stituation sounds like the hitch is not restoring any weight to the front axle of the truck. Could be the hitch is wrong type, could be the hitch is not set up properly, could be the hitch is undersized for the trailer weight. This causes a loss of steering control, and, even subconscious hand movement can cause sway.
Very good chance, you also have "P" rated tires that may be under inflated for the load. This causes sidewall flex in the tires.
Pictures and more accurate weights, would be a huge help.
First thing you should do, take truck / trailer combo to a CAT scale. You want axle weights on truck and trailer with WD bars hooked up, axle weights on truck and trailer hitched up without WD bars, and axle weights on just the truck. The numbers from those three scale passes, will determine actual weight of the truck, actual weight of the trailer, tongue weight of the trailer, and how much weight is being restored to you front axle.