While that truck should be able to tow that 27' trailer, it is just not a very good tow truck. The 145" wheelbase, 20" P-rated tires, 3.31 gears, are all items that take away from your tow capacity. If you're going to keep the truck, then switch your tires to LT tires, at least the rear ones. You can't do anything about the rest of it, at least nothing that would make fiscal sense. Airbags aren't going to help with the sway. I would put a set of Timbrens on it (<$200 and 20-30 minutes of time). A tune for that 5.0 wouldn't give much satisfaction, and will void whatever power train warranty you have left. Like others have said, check your weights and your hitch setup. Other than that, either get used to the sway or get a different truck.