The newer (2015+) F150s have issues towing heavier TTs (ref. https://www.f150forum.com/f82/). Anything over 6000lbs and you have to be very careful with your weights (payload), weight distribution (tongue weight), and the setup of your WDH.
Your TT has a GVWR of 6500lb but it should be OK if you do have the tow package as the GCWR will be at least 14000lbs.
Given your issues started when you fully loaded the TV+TT, I'd start with your payload...have you overloaded the truck? You need to look up your payload rating on the yellow sticker (it will be something around 1600lbs). Assume the hitch weight of the TT is at least 800lb, then you have your payload minus 800lb to account for everything in the truck...driver (minus 150lb), passengers, dogs, beer cooler, BBQ, firewood, etc. If you are over the payload rating, then you need to leave stuff at home or move it into the front storage of the TT. I had this problem and now only carry bulky lightweight items (e.g. a bike) in the bed of the truck.
I'd also check what you put in the TT and where. Yours can carry 2000lb...do you really need to haul a ton in the TT? Make sure you keep sufficient weight in the front of the TT and don't just move things to the back thinking it will help. I experimented with moving just 100lb out of my front storage to behind the axles (I have a rear storage area) and ended up moving it back because it towed better nose heavy.
If your weights are good, then check your WDH set-up. If your dealer installed it, then they could have guessed wrong about the setup for a fully loaded TV and TT (I had this happen when I upgraded from a 6000lb to 7000lb TT the WDH needed to be dialled in with my F150 at full load). If you can, re-install/setup it with your packed TT and a representative load in the truck (ref. Joezeppy's post).
P.S. I don't think air shocks will do anything for you (I'd bet the truck is level)...this is all about weight.