I have the same truck and used to tow my Jayco 27bh with it. Very similar to the weights you listed. IIRC the max trailer weight for the truck is 6k or 6500, not sure. The payload is where the issue is. Between passengers and tongue weight you might be over your payload. I never felt comfortable towing my trailer with this truck as it was right at the edge, if not slightly over, the weight ratings. I stayed local and avoided the highways as much as possible. I put Bilsteins on it, LT tires and ran with higher inflation psi to keep things as solid as possible along with the tranny cooler and temp gauge. All those things may help but do not increase the weight ratings.
I have since upgraded to the truck in my sig.
I still have the aftermarket tranny cooler and temp gauge in the burb, swapped out the Bilsteins and the LTs for a smoother ride since I no longer tow the RV with it.
You should really check your weights to make sure you are within specs of the truck.
The GMT800 trucks are notorious for spongy brakes. I also swapped out the 16 inch wheels for 17s and installed GMT900 front brakes. Larger diameter rotors, bigger calipers and pads. Direct bolt on. Made a huge difference in braking feel. Again, this does not increase towing capacity but did help out with braking on my truck.
Thanks,
Brian
TT = 2018 Wildwood 31KQBTS
TV = 2006 Ford F-350 CC DRW 6.0