We have a 2010 Tundra 4 door double cab 4x4 with the 5.7 and tow a 32' Forrest River Surveyor trailer weighing in somewhere around 9,000#s. I have the ez lift WD hitch and started out using the sway control that came with it but found it handles better without it. We've towed thousands of mikes with this set up, no issues to speak of.
Our last long haul was VA to Tallladega, AL, to Nashville, TN and from there to Martinsville, VA and home. The hills between Nashville and Martinsville were no match for our set up, the Tundra did quite well.
Just before that trip I added Firestone Ride-Rite airbags and was even happier with our towing experience, helped take the shock out of our wonderful interstates.
I'd have to question the 5000# capacity of the hitch on your truck, Are you looking at the numbers on the bumper??? I believe that shows 500lb tongue weight and 5000lb trailer weight. Our Tundra has a class 4 hitch which I believe is rated for 12000lbs and 1200 lb tongue weight with WD hitch, both of which exceed the trucks specs but the hitch was on the truck when we purchased it.
Happy Travels with whatever you decide to get.
2010 Toyota Tundra SR5 TRD double cab 4x4
2015 Jayco Jayflight 32 BHDS
Prodigy Brake Control
Eaz-Lift WD - Eaz-Lift sway control