The GCWR (rating for the combined weight of loaded truck and loaded trailer together) for your truck should be 20,000 lbs with the 3.73 axle ratio, if I remember correctly, unless you have the Tow Boss option, which included the 4.30 axle ratio. With the 4.30 ratio, the GCWR was 26,000, if I remember correctly.
Our '05 E350 van with 6.0 diesel, auto trans and 3.73 ratio was rated 20,000 GCWR. If you have the 4.10 ratio, it could be more, maybe 23,000.
Figure the truck likely weighs 8000 lbs by itself, so if your GCWR is 20,000, your loaded trailer weight should be 12,000 or less.
Your dually has lots of payload capacity and rear axle weight capacity and can handle the hitch weight of any typically loaded 5th wheel trailer that it is rated to pull, and any bumper pull trailer as well.
The 6.0 diesel doesn't have a good reputation for reliability. Some of that is well deserved, while some is over exaggerated. Many 6.0's give no problems. Others are a nightmare. The motor does work awesome when it's running. Ours could pull down a house and not break a sweat. Never had any problems with it. Whatever you do, DON'T CHIP/TUNE IT!!