Yes, go with the 3/4 T for that TT and size of your family.
That hitch wt. is dry wt. and the actual will be more like 1200-1300 lbs, and possibly a bit higher still. The payload in the truck can be upwards of 1K lbs by the time it's loaded with the family, pets, groceries, camping gear, bikes and other stuff. Even if you can find a truck that will handle the weight, the 3/4T trucks still have a stronger frame and bigger brakes AFAIK.
We have an F250 that is a dedicated TV. Our 29' TT is lighter at around 7K lbs loaded for camping. Totally love the 3/4 ton truck (a replacement for our old F150). Headed off camping yesterday with some friends. We swapped trucks for a few miles and I got to drive their 1/2T Chev towing a similar sized TT. We both strongly agreed that the 3/4T truck is definitely noticeably "beefier" and a better towing machine. Personally I didn't like the feel of the 1/2T even though it is well within weight specs. Also, they have friction sway control and ours is a Reese DC. IMO, the DC feels and performs better.
If you get an extended or crew cab, the extra wheelbase will make for better handling and sway control. I you can find 2WD in a 3/4T I would get that unless you really need 4WD. The 4WD is higher off the ground (5" or so) and reduces the payload rating.