Fords over the years have been notorious for wonder. You need to find a real front end alignment shop who knows how to fix the problems, not just throw parts at it and set it to factory specs. Which IMHO are inaccurate. Then and only then you need to take the truck for a road test to be sure it is handling correctly. Then it is definitely time for new tires. Who ever put those underrated tires on did no one any favors.
Now, once those things are correct, it is time to hitch and get to the scales. I would try and find a closed DOT highway scale and avail yourself of it for a long time. You need to get front, rear axles weighed plus the total weight without the trailer. Next the same weights with the trailer attached, with and without the spring bars properly connected.
I totally agree unloading the front end 500 pounds is a big part of your problem. In reality it should be adding some weight. There are a mired of things that could be causing this, from hitch head angle to spring bar tension. Personally if you really want to cure this problem in a hurry and all but eliminate sway in the process, look at a new hitch setup. Something like the Hensley Arrow will help tremendously.
