First, yes, move some weight rear-ward and try that. Can you not put the other bike from the truck bed into the TH? That would ease up on the truck rear axle weight significantly. Typically toy haulers are designed nose heavy to offset the "toy" weight in back.
What are you defining as sway? Some wiggling back n forth when changing lanes or correcting steering, OR uncontrollable sway that doesn't go away until you apply trailer brakes or slow back down to below a certain speed?
If you have E tires, airbags and some weight distribution, IMO the truck should handle it ok (aside from possibly being over hitch load ratings).
