Your truck might be on the verge of it'd gvwr rating with a 7800 pound trailer. It depends upon loading in the truck and trailer. However, it will not show in the systems you described. The symptoms you are describing are typical of weight balance in the trailer. Someone more wise may comment if a trailer tracking problem may show some of these symptoms.
It doesn't matter how light (or heavy) the trailer is if it is balanced. A utility trailer, loaded with boards hanging over the rear too far taught me my lesson. Until then I would never have believed how a vehicle can be thrown around. Redistributed the load, bought new shorts, and I was good to go.
Too little tongue weight is frightening. The swinging trailer could throw you off the road. You should have about 12 % of the trailers weight on the hitch. Protect yourself and measure. Don't use a friction sway bar to try to mask this symptom.
Carrying water in the trailer, or not, is usually not a issue. In many cases the fresh a d grey water tanks are over the trailer axle, and don't impact the balance a lot. 30 gallons of water is about 300 pounds. This 'could' require you to change tongue by about 30 pounds. It just isn't that critical. The only time I would worry about carrying or not carrying water, is if I am loading the trailer to maximum. 300 pounds more in the trailer, when I'm already pulling a big wind break isn't going to affect my gas mileage enough to worry a out.
Did you have full propane tanks, and a battery on the tongue? Do you have anything hanging on the back of bumper?
Why don't you load more weight into the front of the trailer and give it try
Borrow some five gallon water jugs (or someone's set of free weights) and load them into the front and see if it makes a difference.
Solve the loading problem and then worry about sway control