I'm convinced most of the sway people experience with campers is due to insufficient spring rate. You can use progressive springs to make a little better unloaded ride, but need enough spring under load while the suspension is cycling to reduce sway. Torklift products will preload your existing suspension to get the most out of it and is a good start. If you need more help, you need more spring.
I have to agree about air bags amplifying rebound - As long as you use them as helpers instead of primaries, they can help you. The people pointing out that air ride suspensions work fine, need to realize the suspension system and bags are designed for this application and typically have multi-link bars keeping movement and position under control.
Shocks with proper dampening and rebound will help with sway but their principle purpose is to keep rate of movement under control.
Stabilizer (anti-sway) bars are great on roads but an impediment on uneven surfaces. If you look at large trucks design to haul loads off-road, you will not see heavy stabilizers because articulation is your friend for control and traction when not on an even plane.
Although I think my off-road driving can be tamer than some driveway cutouts, driveways are a good place to observe how different vehicles react. You can see the interaction of the suspension and chassis as hey enter and leave the cutout. I tweak and adjust my off-road toy suspension more often, but have crossed some of my experience between toy and truck (yes, I test some of my changes on driveways).