We have your same truck, and tow a Cougar 324rlb that weighs just under 12000 lbs loaded. The loaded pin is +/- 2000 lbs give or take 200 lbs depending on how/what we load. We have larger tires that have a capacity of 3750 lbs each at 80psi. The original tires have a capacity of 3042 lbs, which is how you get your RAWR of 6084 lbs, since the actual rear axle itself is rated at well over 10,000 lbs on these trucks.
We are a little over the truck's GVWR, but effectively under our RAWR with the higher capacity tires. Your truck at 6700 lbs weighs about 400 lbs less than ours so you have even more payload available.
For four years, our truck has pulled and stopped our fiver with nice handling and no issues whatsoever. If you still have the stock shocks on your truck, I would recommend replacing them with Bilstein shocks. They will make a big difference on dips and such.