It really is a lot of trailer for an SUV. Something that size is firmly into 2500 series pickup range. You're nearing the capacity of the tow vehicle if you get anywhere near the mountains.
Now when you say "sway", do you mean it sways, as in gets tail happy and pulls the tow vehicle around, under normal conditions at highway (55-60) speeds with no wind? Or do you mean you get the "suck/push" when you get passed by an 18 wheeler? Some of the Suck/Push is normal, and you learn to anticipate it and compensate for it as a regular part of towing. The rig getting tail happy and dragging your tow vehicle around is not normal, and needs to be addressed immediately.
I'd really like to know the amount of actual tongue weight, and that figure as a percentage of loaded trailer weight before doing anything else. Those numbers may tell us a lot. Our trailers are similar size and weight, and I'm getting zero sway....the main difference is I'm towing with a long wheel based F250. It's an older truck, but the tow rating is closer to 10,500. I also run with full fresh tanks and propane tanks (front) and load all my heavy tools and accessories in the front storage bay. It works great, and I tow in VERY mountainous conditions.....but even I'm looking at a SRW Ram 3500 Cummins diesel for my next truck.
The closer you get to your tow vehicle's capacity, the less fun towing is.