Agree with Lynnmor, a 5K loaded trailer wt will be about it due to low payload of 1,070. At 5K, you will have 600+ lbs tongue wt. That only leaves about 400 lbs for everything else in the truck.
Your engine/driveline is designed to tow 10,000, but with a GVWR of only 6,800, you simply don't have the payload.
Similar trucks may have 1,520 payload, but the sticker on YOUR truck indicates 1,070 payload as it left the factory, before going over GVWR. This has to do with options added to YOUR TRUCK, options add weight.
Some choose to go over GVWR a little, but stay under RAWR of 3,900.
Hope you find a nice trailer, that works for you!
Jerry