Add tongue weight up to 12% or so, add a weight distributing hitch, level out the trailer, trailer tires at sidewall maximum and it should handle fine. Not sure how you can tell it's swaying behind the trailer axles and not in front of them - if it's swaying the whole thing is swaying, it's not a two part deal.
Lots of people with one ton trucks like the F350 think they don't need a weight distribution hitch, but the maximum tongue weight without WD is only 500 - 600 lbs depending on year. You will need to have 900 - 1000 lbs of tongue weight, so size the hitch you buy accordingly. I would buy a 1000 - 1200 lb hitch for your application.
Brian