In my opinion you have plenty of power with the 5.3L/6 speed combo to tow a sizeable trailer, as mentioned you are limited on the payload.
When you mention two little ones, is that all there will be? My 05 1/2 ton Suburban was 200# over it's GVWR with six of us and two dogs with our Passport (820# of tongue weight).
If you go with one of the trailers APT mentioned and it is really only four of you, you should be able to manage staying in your limits.
We really like our current TT. With the deep slide there is tons of room for six of us and each kid has their own stationary bed.