Very simply put. You need more truck. Or you need a much lighter TT. Your choice. But you ain't got enough truck for that trailer. and remember. the weight of the passengers, and cargo in the truck reduces the towing capacity.
I also don't believe you are taking the pay load into account. 7000lb GVWR is going to leave you with a payload in the 1400lb range. A TT that starts with a 900lb tongue weight will have a 1200lb tongue weight very quickly. Guess what. your payload is gone. you don't have any left for the family, and hitch.
In short. IF you take 2 vehicles, don't load anything into the truck, except YOU. and load very little into the TT. You might have a small chance of staying in your Truck's capacities. A very small chance
You need more truck, and NO 3.42 is NOT a towing gear. Much too high. And while you are at it. You better check the hitch capacity. 1200lb, Even 1000lbs may be heavier that a 1500lb stock hitch can carry
