I believe those who say the dually feels more stable... but I also believe... in most situations... the difference felt is because of the heavier suspension on the dually not because of wider stance and 4 rear tires.
Some of the newer one ton single rear wheel trucks have about the same payload of the one ton duallys eight or ten years ago. For that reason some of the heavier pin weights might work with a newer one ton srw payload. However, in some ways that is not really about stability, it is about payload weights. Only you can determine what is best for you by driving both in as close to identical trucks as you can find. I am with those that still think the stronger axles, wheels and tires are also a safety feature and that fifth wheels that weigh more than around 16,000 pounds are likely to give a better tow with a dually. At 16,000 pounds the pin weight will be at least 3200 pounds, the hitch another couple hundred and all the folks, dogs, tools, etc, in the truck bring the total to 4000 or more pounds. Do you absolutely need a one ton dually to tow that weight? Probably not. But a dually will make the tow nicer. The dually option is probably the best and perhaps least expensive option available for carrying more weight as it's cost is very minimal compared to the price of the already expensive truck. In some cases the dually option may be less than finding you need to upgrade the suspension with bags and stronger wheels and tires to handle the weight on a single rear wheel truck. Only you can determine what you need and stability is in many ways subjective. You have received some excellent answers on this thread but it is time for you to go drive both.