Most common causes of trailer sway (list not in any particular order). Trailer sway can be caused by any one, or a combination of two or more. Many are a simple check / fix.
1. Insufficient tongue weight. Must be minimum of ten percent of total trailer weight.
2. Towing trailer with nose up attitude. Trailer nose should be level to slightly down attitude. Maybe your hitch ball is too high.
3. Insufficient weight loss restoration on truck's front axle. When you hang the tongue weight on the hitch, the effects on the truck are like a see-saw, it adds weight to rear axle, takes weight off the front axle, and magnifies the pivot point at the hitch ball. Causes - weight distribution hitch is not rated for your tongue weight, or it is not adjusted correctly. If there is too much weight missing from front truck axles, even subconscious hand movement (could be simple as breathing) causes the trailer to wiggle on that magnified pivot point.
4. Truck is loaded beyond it's payload / GVWR. Look at your tire / loading sticker (on drivers door jamb). It will have a number for "max occupant / cargo weight". That is your truck's capacity to carry the combined weight of everything (including aftermarket accessories (bed cover / caps, bed liners, step bars, etc), added hitch / sway equipment and trailer tongue weight) and everybody that was not in it, when it left the factory.
5. Bad roads.
6. Tire sidewall flexing. If you've got (P) passenger rated tires, they are known for having softer sidewalls than a (LT) light truck tire.
7. Unbalanced / under inflated tires on truck, trailer, or both.
8. Wind.
9. Bent / misaligned trailer axles. This one is rare.
Looking at your numbers (tow rating and trailer's dry weight), I suspect a weight issue. That "max tow weight rating" was calculated without passengers and cargo. They're calculation included a driver, weight distribution hitch, and the estimated tongue weight from a 7600 lb trailer, would use up all the truck's payload. If you add passengers and cargo, the truck no longer has payload to carry the tongue weight of a 7600 lb trailer and the towing capacity is reduced. Save your money, air bags won't fix that. If you've got cargo in the truck, it may help to move that to the trailer, preferably above or slightly forward of the axles.
Average trailer load (dishes, pots and pans, camp chairs, clothes, bedding, BBQ equipment, groceries, water, etc) weighs 800 to 1000 lbs. Your trailer weighed 6300 when it left the factory, probably 6500 when it left the dealers lot, and fully loaded, it could weigh 75 - 7600 lbs.
Tongue weight is NOT a constant number, it goes up and down during every trip. It averages 12 - 13 percent of loaded trailer weight, but can be higher. Depending on location (in relation to trailer axles), holding tanks can have a significant impact on tongue weight. I have a trailer (8300 loaded) with black / grey tanks above the axles, fresh tank behind the axles, and galley tank up front. My tongue weight can be anywhere between 975 and 1225 lbs. All depends on fluid levels in the tanks. Percentage wise I run 11.5 to 14.5 during any particular trip.
True towing capacity is limited to the weakest link in the truck's overall ratings (GVWR, GCVWR, Payload, Tow Rating, axle weight, tire weight, and hitch weight). Most often the weak link is payload.
Note: Your uncomfortable feeling could just be caused by being on the edge of the truck's capabilities. I've been there. I watched the weather very closely. Windy days were unpleasant, and if the weather report said breezy, I stayed off the road.