Scary Handling Problem

OK, time to get nerdy here. Your lateral cycles issue is from you exciting (not your fault) a natural "mode" (or frequency) that is inherent to your combined vehicle setup. That is what is giving you the white knuckle ride until you finally dampen it out. While I cannot give you the solution to the reluctance to turn what you need to do to kill off the lateral oscillations is to change the natural frequency or seriously limit the lateral movement of your RV/toad combination when it tries to do its dance. In my opinion the single best way would be the rear track bar. This will locate the rear axle more positively than just being held by the leaf springs. This becomes more of an issue at gross weight increases and would me exacerbated with something acting on the moment arm (long lever) that is your rear overhang. The addition of the track bar will limit movement and should increase the natural frequency of the axle/chassis interface.

How's that for a mouth full.