Much good advice above. In a nutshell, the goal is 10-15% of trailer weight to be on the tongue. That balance of the trailer alone is critical. Next, you must ensure you are transferring enough weight to the front axle with the WDH. This requires the right spring bars for the weights. Check with Reese. Then, the Ram air suspension requires you to follow the manual for spring bar setup. You drive with suspension in Normal. It is true that your trailer has a lot of surface area and this provides a challenge to 1/2 ton truck tires. The stiffer sidewall of E range is superior but very stiff for the unloaded 1500.
Rather than change the truck, you can eliminate this sway by using a ProPride hitch, or the Hensley Arrow. These eliminated sway completely. I have thousands of miles towing Airstream trailers (28-31 feet) with a large SUV and 3/4 ton diesel pickups. The first thing I would purchase if I decided to tow an Airstream again with my Ram 1500 Hemi E-Torque would be a Propride hitch. You will not notice passing Semi's even in strong crosswinds. There is nothing like these, and worth every penny in my opinion. I prefer the updated Propride design to the Hensley.