Here's the best info I could find on a 2003 1500. I don't remember if it's a reg cab or quad cab. Regardless the payload numbers are about the same 1540-1570lbs.
2003 RAM Pickup Payload and Towing Charts
Going with the 1540lbs would be practical. That's the number form Rams site so historically it will be lower when delivered.
So figure 1500lbs of payload.
If you're looking at a 5975 dry weight TT, then is that from a brochure? I'm pretty sure you're giving brochure dry weights because you also list the tongue weight and the tongue weight is never listed anywhere except in the brochure. Those weights are not accurate. You need the dry shipping weight from the TT's sticker. All TT/5'ers are required to have shipping weight stickers on them when delivered to the dealer.
Also the dry tongue weight is not right either. It's under the 10% min for towing. 12% I usually what it ends up being.
Just for figuring purposes lets say the TT gets shipped at 6200lbs. I don't remember if you're just camping for fun or seriously traveling for months at a time. As a general rule you will pack close to 1000lbs with you. You may not think that and we didn't either with our 1st TT. But posts and pans, cooking utensils, bar b que, clothes, pantry items, food in the fridge, cleaning supplies, towels, some tools, the WDH, wood blocks for leveling, jack for flats, lawn chairs, sewer supplies, axe for firewood, etc. No way you can just put 500lbs in.
Believe it or not if the dry weight is 6200lbs then it will weigh 7200lbs loaded.
Now that puts the tongue weight between 720 and 1080lbs. Split the difference and now you have 900lbs on the tongue weight. That 574lb brochure weight doesn't include propane, a battery and anything you put in the front of the TT.
1500lb payload for the truck minus 900lbs for the tongue weight and you're at 600lbs left for passengers and whatever else in the truck.
Max tow is 7850lbs. The truck has a 13,000 GCVWR.
Empty truck 5081
That leaves you with 719lbs for passengers and whatever else you put in the truck. You'll notice that I'm not figuring in the weight of the truck as shipped, that's the dry weight from the Ram site. It doesn't differentiate between trim levels or things like running boards or other add-ons.
IMO you really need to load up the truck and get it weighed to see how much payload you really have. Or better yet quit looking at 6000lb dry brochure weight trailer and look at 5000lb dry shipping weight stckerd TT's. 7850lbs of tow rating is pretty low.
Another thing to cinsider is the 03 5.7 Hemi is not as good as the newer Hemis.
Those rpms are way up there so the heavier you tow the more the hemi will be working. My guess is it will be revving pretty good under normal driving.
Looks to me like you will be close to max on everything.