Your numbers are correct. What your missing is that most everyone carrying a TC is over their GVWR and has modified the truck in some way to help handle it.
You say you weighed the truck and it was 7662? How did you arrive at that weight. I've never seen a truck scale that weighed in less than 20 pound increments, most of them weigh in 100 pound increments.
Did you remove the tailgate before you weighed it. That will come off with a TC so that helps some.
For what it's worth my '05 Chevy dually has a GVWR of 11,400 but I weigh in at about 13,500 with the family, camper, and boat hitched on behind. I don't have stock tires, shocks, or hitch and I added air bags.