PULLING 5TH WHEEL
I have essentially the same truck from 2001. There are 2 questions, how much can I tow safely, and how much should I tow?
The first step is to look in your owner's manual. It should tell you, for your model truck (long/short bed, regular/extended cab, 2WD/4WD) and gear ratio, how much trailer you CAN tow with that 6.0L engine. 10300 sounds about right (my manual goes from about 9900 to 10600 for the various 2500HD models with 6.0 engine and 4.10 gears). This assumes, of course, that your truck is empty, with driver and 1 passenger. Anything added to the truck (accessories, including the hitch - 100-200lbs) or carried in the truck (including big or extra people), is subtracted from the trailer capacity.
What you really want is the GCWR (Gross Combined Weigh Rating) for your truck. Knock your dealer down and sit on him until he gives it to you (you may also be able to find it on the website). This will tell you the total acceptable weight of the truck plus trailer. Take the truck, loaded for travel and with all occupants, to a scale, subtract that value from the GCVW, and you know exactly how much trailer you can safely pull. Or you can guess, perhaps 8000 pounds would seem to be a safe value in most cases, possibly more if you never carry anything in your truck.
So the truck is a great towing truck, but I'm afraid that the 6.0L engine has trouble getting my 6000 lb trailer up any hill or grade. It will do it, just not quickly. If you are not in a hurry, you can crowd your GCWR - loaded truck weight value, but if you want to maintain highway speed, you'll need either a VERY light trailer (maybe a popup

or a different engine (BIG gas or most diesels).