Your current truck is carrying 8,300 pounds on the axles. Assuming a typicall 2500 truck would be somewhat heavier even empty you might be around 9,000 to 10,000 with the same ATV and trailer on a 2500 truck. Most of the newer ones have a GVWR of around 9,200 to 10,000 so it should just be fine. 2500 trucks 10 to 15 years ago typically had a GVWR around 8,600 to 9,200 so most of them would likely still be overweight.
No, a dually isn't required but it would be wise to step up to a 3500 single rear wheel truck.
As for replacing your rear axle, that doesn't address springs, frame, brakes, or various other things that are beyond their specification with your current truck. You likely couldn't get a 1 ton rear to fit anyway. The spring perches will probably be a different size, the U-joint yoke a different size, etc.
A 3500 truck is simply more massive than a 1500 in every way: frame, engine, drivetrain, axles, driveshafts, brakes, rims, tires, etc. Your driving experience would be night and day different.