We made the trip home with 35 psi in the airbags. The sway was greatly reduced. I might try it with the bare minimum of air in the bags. The overloads are not in contact with the bump stops, and I'm not convinced they need to be at all times.
Thanks to everyone for chiming in with your knowledge and experience. I think I understand how all this stuff is supposed to work now, which will help me to figure out the best configuration for my set-up. I think the first thing I might add at this point is new sway bar bushings.