You came from a heavy, stiffly sprung 3/4 ton truck, to a light, squishy (especially by comparison), air ride half-ton. There's going to be some difference. The DT Rams tow great, for a half ton, and the ride is amazing loaded or unloaded. But, in my experience, coming from a Tundra (also on the stiffer side, a little heavier, and kinda a half built like a 3/4 ton), I also noticed the Ram felt a little less stable pulling my car trailer setup, which I've done hundreds of times with the Tundra, and even pulled the exact same vehicle once with the Ram. My suspicion, is to upgrade the anti-roll bars, as I assume they are hollow and relatively thin-walled to save weight and help with the ride compliance. I'd trade just a dash of ride quality for that extra stability when hitched up.
Now, here's what I'm not saying: The Ram is by NO MEANS unsafe, unsettling, nerve wracking, or dangerous feeling to tow with. I'm only noting a small difference between two trucks that I had / have, and got the opportunity to tow the same trailer with the same load on the same roads with. If your experience with the Ram invokes words such as above, especially when unloaded, then I definitely suggest a trip to the service department asap. Trailer setup is also key, and if you haven't adjusted the hitch / weight bars from their position that you used on the Avalanche, I'd suggest starting there and setting the trailer up properly for THIS truck, using the owners manual as a guide, as there is a specific section about air ride and wdh's. Edit: Just remembered you were talking about boat trailers, so probably not applicable in this scenario, but you get what I'm saying.