I consider "floating" and up or down motion and "tail wagging the dog" a more side to side issue. Do you have issues at times, due to wind or other passing vehicles, where it doesn't want to stay in the middle of the lane and you are constantly correcting?
Considering the weight of the trailer, I'm sure you will be happier with 3/4 ton vehicle. A longer wheelbase 1/2 ton with the appropriate payload (which is not always easy to find) would likely be an improvement as well. As for your Expedition, I'm not sure why the air suspension would be the issue, it is just a spring that supports the weight vertically. The shocks control the motion vertically. The independent rear suspension, if in good condition, should locate the rear wheels side to side very well. I have heard of issues with the Expeditions where the rear suspension bushings wear out and get sloppy, another potential issue. How many miles are on it?