Looks like a F250 would be a better truck to be towing with. Don't think you are looking to purchase a newer truck.
I would tow with all tanks empty and fill the water tank when you get to your destination if needed. Dump all tanks before you leave the campground.
I was in a similar situation with a 2005 Silverado 3500HD SRW with a diesel. Was towing at least 900 to 1000 lbs over the GVWR. I traded the truck for a 2011 when the trucks were beefed up and I am now towing within the GVWR of the truck and worry less.