On a one time basis, I will be towing a 32' travel trailer. The trailer is totally empty and dry and weighs about 4,500 lbs. There will be about 200 lbs of cargo put inside near the front, so toungue ...
The secret to no sway is plenty of tongue weight. You have plenty so do not worry about it swaying. When in doubt I give it test by wiggling it at 45 MPH and see if it gets worse. If it does get on the manual trailer brake and gas it until it stops and then slow down with the trailer brakes. Shift something further forward.