Calling it sway is a bit inaccurate, it covers two different trailer actions. One is the tail wagging, the other the trailer rocking side to side. Which is more accurately called sway. Both actions will affect the tow vehicle's steering if it has inadequate rear suspension.
When I pulled my 4x8 utility trailer loaded with quad and 3 jerry cans across the back,with the Jeep Cherokee things got interesting at highway speeds, I had to slow down to stop the tail wagging. However with the same poorly loaded trailer, I didn't notice the misbehaving on the 1ton.
From my experiences, I would say most, if not all trailers have well enough designed loading that sway, both kinds, that cause unwelcome steering inputs are a fault of an inadequate rear suspension transferring trailer actions.
