Ahh! I already said ignore tow ratings for RVing, 8000, 10000, or 12000 pounds does not matter for your truck. You stated you have 1380 pounds of payload. Is that what the yellow Tire and Loading sticker on the truck you own has, as I showed above? You are likely to exceed that well under 8000 trailer weight with a family and some camping gear in the bed. Do you have a bed cover or truck cap too? It does not include any people, no 150 pound driver, only all fluids. Tow rating usually includes a single 150 pounds driver only.
With a dry weight of 7000 pounds and a family expect you be 8000-8500 pounds loaded for a long weekend trip. That means 1000 pounds of tongue weight. A weight distributing hitch does not reduce the tongue weight, but it does reduce the weight on the truck's axles. It distributes weight from the TV rear axles to both the TV front axle and trailer axles. The net result is about 25%. So, 1380-750 of TW pounds leaves 630 pounds for the family and cargo in the bed before exceeding the truck's GVWR. That will be tight for a family or 5.
You got what you got and any changes now are going to cost dearly. Many people tow overloaded by 500-1000 pounds and they don't know it or care to. Get a good WD hitch with integrated sway control (Reese dual cam, Equal-i-zer 4-pt) and drive accordingly. I'm sure you'll find the Tundra does very well towing that RV with a good hitched that is adjusted correctly.
