Post the payload number as requested. Your tongue weight is going to be around 1000 + lbs. A good weight distribution hitch does not increase payload, nor do air bags. IMO, I would not tow that monster TT with my family in your truck.
You may go ahead and do it though - and that's your choice. Good luck on traveling the country in that combo. And, I'm sorry, the numbers matter before you get your towing experience. Either have a big enough cushion if calculating (guessing) or packing up the TV as you would to go on a trip and getting it weighed (knowing). Then getting the appropriate TT.
Google Cat scale locator to find one near you.
Best advice I can give is to find out how to figure it out from people who know on here, then actually do it. Not just listening to "its doable" comments.
I'm sorry you got sucked into the fantasy world of tow capacity. That number is determined by pulling a flat trailer with weight. When your pulling a big sail down the road - high walled TT, things are a lot different.
Good luck