2010 Dodge Ram 2500 Specs
Ok, I found the GVWR of your truck right away. I will keep looking for your Combined weight and max axles loads. Remember the curb weight of your truck is the GVWR minus your cargo capacity.
The Gross weight of your truck (the maximum IT can weigh with you, your family, your hitch, your pin weight, and all your gear is 8800 pounds.
From your door sticker it says your max cargo load is 2244 pounds.
Subtracting 2244 from 8800 should give you your curb weight with the diesel engine or 6556 pounds. That is JUST you (provided you weigh 150 pounds) and a tank of gas.
So working with the 2244 pounds of "Load"
Subtract from 2244 the weight of your hitch - maybe 144 pounds
Subtract from 2100 the weight of wife and family 300 pounds
Subtract from 1800 pounds junk in truck like cooler with beer 100 pounds
Leaves 1700 pounds available for pin weight.
Looking at your numbers, it shows you seriously overloaded. Time to trim back on what you are taking. If no camper you weigh 8200 pounds then you only have 600 pounds available for the camper. As you sit, your truck is weighing 10,620 pounds with the camper on its breaking back.
10,620 - 8800 = 1820 pounds overweight on the truck.
Your trailer as currently loaded weighs 20,560 - 8180 = 12,380 pounds which is under its max of 13805.
Your pin weight is 10,620 (after camper) - 8180 (before camper) = 2440 pounds.
Your problem is pin weight on the truck. Now, having said that, many will tell you your truck can handle it. Maybe they are right, but Dodge seems to think not for some reason.