Can you send a pic of your set up, close up of your hitch set up and one of the truck attached. I am thinking when you say stabilizer bars you mean a Weight Distribution hitch and the sway mechanism is a friction style sway bar.
With out the pictures what I am about to say is a guess and my opinion only. Trailer is 6350 dry so loaded up lets say 7400 with a few perks and your gear. That would give you a hitch weight of around 962pds which is taking a hefty bite out of your 1425pd payload. Means you only 500pds or so left for the passengers, any gear you chuck in it, the hitch and fuel. Not the end of the world from my stand point but some will tell you it is.
Before you hook up the trailer measure from the ground to the bottom of the lip of the front fender and copy it down. Now connect your camper and hook up the WD, take the measurement over again and if you are within an inch you have transferred sufficient weight from the back axle to the front axle. If you are more then an inch off try adding another link to the WD and see if that does it. If you cannot get it to within an inch it is time for a set of Air-bags.
I tow a dry camper of 6500pds, loaded up 7600pds, 988pd hitch weight. I have a Reese dual cam anti sway weight distribution set up and have no issues getting the front within a inch of its original ride height. The big thing comes down to how it tows, do you have any of the "tail wagging the dog" effect where you feel tugs from left to right toward the back of the truck? This can be common with rams and heavier loads as the springs which provide us with a much better ride have a little more give and can cause some back and forth motion. Weight distribution should help eliminate this but if your not getting it back to a sweet spot it can cause a problem. As well the tires play a roll, pardon the pun, with that kind of weight in the back your tires should be a LT tire and most likly are. When I replaced mine I decided to go with an E rated tire (10ply), stronger side wall then most 4 and 6 plys which reduce the roll effect that take place from the tires.
In the end, the airbags are not going to hurt anything! I just bought a set for my truck, airlift 1000's. I bought them not because I feel I needed them for towing but for those occasional loads of river rock or lumber that makes the truck squat bad. With that there is no WD that allows me to transfer weight from the back to the front and its not good for the rig. The airlift 1000 kit is cheap, great warranty and was told by many very simple to install.
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you will find tons of similar post with your question.