Towing capacity is based on your whole package (engine/trans/axle), and a big part is wheel base. Your truck has a max towing capacity determined by FCA (with the help of NHTSA and DOT) depending on your configuration. That said, swapping 3.21 gears for 3.92 gears will make towing the same load easier for the engine/trans and may theoretically bump up your capacity slightly. However, it's on the order of only 500-1000 pounds depending on the gear change and any other changes and your starting configuration. If you plan to tow regularly, and the amount you have to tow would be full capacity, then at that point you should just upgrade to a 2500 or 3500.
Still, only towing 6 or 7 times a year, if you are at 6000-6800 pounds, that's a strong case to go up to an HD. If you are mostly towing on flat surfaces and you take it easy, your 1500 as equipped should perform ok.