You've gotten some good advice already. Yes, the heavier the truck, the less you have to "fuss" over sway bars and weight distribution. My F350 could tow our TT like it was on rails with no anti sway gear or WD gear. Hitch the trailer to my wife's Excursion and everything (air bags, WD, sway bars, load distribution) has to be perfect for it to tow properly, even with upgraded anti sway bars and shocks on the truck!
With regard to your questions: look into upgraded shocks (maybe bilstein) and anti-sway bars (helwig). However, if it is in the budget, absolutely trade for a heavier truck.