A 3700 pound "dry" camper will likely weigh, let's say 4500 ready to camp. A typical 13% tongue weight will result in approx 600 pounds of tongue weight.
On that truck, my bet is that will result in a reduction of around 300 pounds from the steer axle. The primary reason we use a WD hitch is to restore part or all of the front axle weight.
Until you take it to the scales, all the rest of this is guessing. My opinion is that yes, a WD hitch will make this a safer and better towing and riding experience. I would use one.
Here is a good reference with some useful pointers:
http://www.etrailer.com/faq-weightdistribution.aspx