Blackcows,  your answer is Yes. GVW is what your truck CAN haul.

I kinda suspect your curb weight of 2006 F350 Dually is more like 8,400 to 8,600 lbs. At least the diesel Crew Cabs 4x4 run in that range. If yours is gas engine, 4x2 regular cab maybe it's that light.  I know my 2006 when I had it was close and my  current 2011 SRW CC LWB 4x4 is 8400 lbs empty.  13,000lb - 8,600 =4,400lbs of payload.  Run it across a scale empty to see what you really weigh. And then make sure your trailers pin weight is not more than what payload you capacity you have left.

Typically you can license a vehicle up to any weight you choose, as long as it does not exceed the GVW and the GCVW  And you see those decals on the side of the truck with the Weights listed.  So if you your truck GCVW is rated at 23,500lbs and it weighs 8,600lbs then your trailer can weigh 14,900.  If 4,000lb of that weight is on the Pin, then 10,900 lbs can be on the trailer axles. Or any mix and match of weights as long as you don't exceed the trucks axles rating, the trailers axle rating or the combined axle rating.

Any DOT check will be with scales. Either at a Port of Entry or a portable scale on the road side.  Different states will limit certain roads to certain weights. Because of bridges or weight bearing material in the road structure.  You often will see Frost laws that allow big rigs to run heavier during the period that the ground is frozen and then be much lighter during the thaw period when the roads are subject to breaking up.  during those periods they actually look at the surface foot print of the tire and how much pressure per square inch of road contact.  For big rigs rated at 80,000lbs, this often means running at 60,000 lbs because they don't have enough axles to spread the weight on the road surface during thaw periods vs what the trucks GCVW rating is.  There are a lot of variables to be  considered.