I have the Ram 2500 Laramie 6.7 diesel. Pretty sure no air shocks. Trailer is 2,540 tongue wt and 14,940 gross wt dry. I load gear such that tongue wt is less, but I don't have much gear.
Ok, so that might be the problem also... 5th wheel trailers need to have minimum 15% of the weight on the hitch; less than that and they will be "tail heavy" which leads to sway. 20% is a more typical amount when loaded, and some rigs may even be as much as 25%.
Also, if your rating is 14,940 empty, what is it loaded? Even with light gear, you could easily weigh 16,000lbs+, which would mean 3200lbs+ on the hitch. Many trailers come from the factory above their rated weight, and they don't include some things that the dealer adds, such as batteries. Again, I highly recommend a trip to a CAT scale to see what you actually have.
But, what is the payload rating for your truck? My 2016 2500 Ltd had a payload of ~2100lbs. For a Laramie, depending on options, you may be a couple hundred more but I doubt you are more than 2500lbs.
So, think about the problems we have here, all that contribute to sway:
-the hitch weight of the trailer will exceed the payload of the truck, likely by up to a 1000lbs or more (once you get fuel and people on board)
-you don't have air suspension (you'd know it by the ALT trailer height button if you did), so we are likely squatting the truck when hitched
-you cannot move the load to the rear of the trailer to reduce the hitch weight or the 5th becomes tail heavy and will wobble all over
Could this be fixed? Not really. Adding air bags and an upgraded sway bar may make the rig more stable when driving. Maybe even you could get it stable enough that you would drive it. But, you will be grossly over the payload of your truck, and especially way over on the rear axle weights. Eventually, something will fail, perhaps dangerously so.
My prior rig was a 2016 2500 Ltd cummins and factory air suspension, with a 35' trailer that weighed ~10k empty and less than 12k loaded. I felt I was pushing it at times and had occasional sway on bad roads or heavy cross winds.
Our new rig is a 40' trailer that is less than 14k empty and ~16k loaded. We upgraded the truck to a 2020 3500 SRW with a 3650lb payload. I would never think of towing this 5th with my 2500, so that's why the truck had to be upgraded.
Your 5th wheel is in a whole new weight bracket. A 3500 SRW may do the job, but a CAT scale may show you that a dually is the right tool for the job. I think the horrible sway you've already experienced is the warning signal that you are vastly "under trucked" for your trailer. I would not ignore the warnings, it could end badly.
All the best in sorting this out.
Brad