I'm going to guess that part of the problem is nose weight on the truck and the other is trailer axles. We used to have a race car trailer that felt the same way. Biggest problem was the camber on the axles in the trailer. I agree with taking it to a good trailer shop and having it checked.
One question though. Does the nose of the truck feel lighter when the trailer is hooked up than without it? You should be able to tell that as soon as you pull out of the driveway.
One thing I want to post is about shocks. People seem to think that the help support the vehicle. Shocks are just a timer as to how fast your suspension travels. Yes, a gas shock does have pressure but it's not there to support any weight. The gas is only to help the performance of the shock.
Good luck and keep us posted. I've been fortunate enough to never have a trailer sway issue myself.
