You have a 3500 with a trailer that should have about 900-1200 # tongue weight. Have you tried without the load bars? Do you even need them? That truck should be able to handle the weight.
Before anyone flames this post, I'm asking. Not telling. Some guys don't use them with larger trucks...
I noticed when I changed my Leveling setup to be heavier in the rear, the trailer didn't handle as well. I never got any sway, but I have an Equalizer and my Sport Trac has sway control, which seems to really work well, so I don't get any sway with a #6000 trailer, and I know my tongue is heavier than it should be as I weighed it with a Load cell. I tow a big trailer with a little truck (V8) and it is the best tow setup I have driven, ever.
I replaced my rear shocks with Rancho adjustables, and that made a big difference to my truck's rear coil springs ability to not sag under the tongue weight. You may want to try to increase the airbag pressure too. I had rear airbags (no springs) in my GMC motorhome and it handled better with more pressure back there. You may try adjustable shocks too. I'd bet some of your squishiness is those tires and the spring rate in the rear.
Edit:
I just reread your post, and My Father in Law had a similar Kit, Year, length, slideout, etc. That trailer had a lot of flex in the frame. It may be your trailer is doing a lot of flexing down the road and screwing around with the axle loading and the wheel camber/caster in a way that is causing it to sway. If the trailer is acting like a big spring, it will start to undulate and that may be what you are feeling.
No offense, but I thought the thing towed like a wet sponge whenever I towed it with his 2006 2500 Chevy. He didn't care for it either, traded it for an Airstream.
