As a F150 5.0, 3.73 owner. I will say, that with a 5.0, 3.73 rear, Your truck will be great with that TT. However. After searching for a long time. I can honestly say FEW F150s, with a 5.0, and 3.73 rears exist.
SO, for us to help, We need more info. Again, what gear, what GVWR package, They as mentioned start in the 6500lb range and go right up close to 8000lb.
As for sway. My TT is in fact 31' long, and weighs 6000lb wet, give or take 200lb depending on where it is going. I was real careful setting up the hitch. I have -o- / ZERO sway.
In other words. The TT follows along like a little lamb. I've never been able to out run it, and it has never tried to pass me. It doesn't try to push me down the mountains, or thru intersections. The factory brake controller works as advertised, As good or better than the Prodigy I had in my last truck.
I can highly recommend the F150 with the 5.0, and 3.73 rears. However. IF the 150 you are looing at ahs a 3.15, a 3.08, or a 3.31. I'd leave it.