Weigh the trailer when you pick it up. Weigh the truck after you get it ready to tow. Then you'll know how much you are over the weight ratings. When you load the trailer, if possible, put everything behind the axles. It may not do much but it should take some of the weight off the hitch. Also, if your water tank is behind the axle, fill it with water. That will increase the weight of the trailer but will also shift weight to the rear and hopefully off the hitch. Then hook your trailer up to your truck and take it to the scales. Weigh the front and rear axle of the truck and you'll know how much the truck is over weight and how much the rear axle is over weight. May not be as bad as you think. If you think you have a problem with over loading your tires, consider larger, higher rated tires. When your done, if your uncomfortable with the numbers, take your truck to a knowledgable hitch guy and ask what you can do.
My 2005 Ford F250 was similar to yours but had a long bed. Dry weight with me and partial load of fuel was 7,600 lbs. Gross was 10,000lbs. So net carrying capacity was 2,600 lbs. The CAT Scale weights were steer axle 4660 and the drive axle 2940. The combined gross axle weight rating according to Fleet.Ford.com is 11,200 to 12,000 lbs. The rear axle gross rating is 6,100 lbs. So in my situation, theoretically I could carry 3,500 lbs in the box before exceeding the rear axle gross weight rating. I'd definitely be over the gross vehicle weight rating but I doubt the springs would break and the tires would melt. The tires on my F250 were rated at 3,100 lbs a piece. That's 12,400 for the truck and 6,200 on each axle.
Now I'm not advocating you drive with weight significantly exceeding the ratings, but I'm not convinced you would have major problems. Don't know about GMC/Chevy, but there's not much difference between the F250 and F350. Another leaf and larger tires and maybe a different axle. But according to Ford, Gross Axle Weight Rating is determined by the rated capacity of the minimum component of the axle system (axle, computer- selected springs, wheels, tires) of a specific vehicle. To me that says if the tires are rated at 3100 lbs each, then the rear axle and all components are rated for at least 6,200 lbs. Some of the folks don't care much for my logic but I've given this subject a great deal of thought and there are tons of RVers out
their pulling trailers that are technically too heavy. Maybe there are some stats that prove what some are saying, but I haven't found any. Another thing I suspect is the 10,000 GVWR of these trucks is based more on licensing laws than the actual weight carrying ability of the truck. In some states, a truck over 10,000 lbs is considered commercial and thus taxed at a higher rate.