With your family of 6 you need to be looking at trailer with a GVWR under 7200# and to be in all of your weight limits you need to be looking at something under 6400# GVWR.
Four years ago we moved up from a hybrid TT to a 7200# GVWR TT (trailer in my signature). At that time I towed with an 05 Suburban with the 5.3L/3.42 axle. From a power stand point I thought it was fine. I added a tranny temp gauge and didn't see anything alarming there even towing on a 4th of July weekend with outside temps pushing 100F.
However, I had scaled my set up and I was 800# over my GCWR (of 13,000#) and something like 200# over my rear axle weight rating. I didn't tow our trailer with that truck long enough to know what wears out first on a loaded truck like that.
Good luck with your searching.