Thanks for all the good suggestions, everyone! That is why I post here. This is what I found out today:
I took my new rig to a CAT scale and got these weights: Trailer - 6040 lbs (max 7250, truck capable of towing 10320), Truck + Trailer - 11740 (GCWR 15950), TW 640. The rig is almost fully loaded and ready to camp - except for some clothes and food.
I lowered the hitch ball one notch as many suggested, after reading the stickies on the forum and doing the measurements. It did seem to tow a little better, but still more sway than I'd like. It doesn't sway into ever-widening arcs, but I can definitely feel every car that passes - my old 18-footer was much more stable.
My next step will be to redistribute some weight in order to increase TW more towards 15%. I have to be careful though, as my current hitch has a TW limit of 750 lbs. The ultimate answer seems like it may be a new hitch that has a higher max TW if I can't get a comfortable ride with this one without exceeding 750.
Thanks again for your help here - there were some very useful suggestions. If there's anything else I should be aware of, please message me (I think the Moderator is going to close this thread now).