Length of travel time has nothing to do with how big a trailer you need. Most single axle trailers will probably have a 3,500 pound axle which wouldn't be enough for a boat and trailer weight of 4,200 pounds. Tandem axle trailers pull a lot nicer anyway.
It's very difficult to get much tongue weight on a boat trailer since all the weight of a boat is in the very back. I towed a 20' boat on a tandem trailer at about 4,500 pounds for many years with about 200 pounds of tongue weight. I towed with 1/2 ton SUV and pickup as well as duallys. I never had any sway or bounce issue.
I suspect you have P tires on your tow vehicle when you should have LT. You probably at least need more air in your tires.
How much weight are you already carrying in the tow vehicle? People, pets, coolers, gear, etc. all add up and reduce the payload of the vehicle. With the additional tongue weight you may simply be overloading the vehicle.