What is dragging? Is it the tail of the trailer? Is it the hitch?
Either way, I think adding a wheel is going to make the problem worse. The problem is that the wheel, itself, lowers the contact point at least another 2 inches, increasing the pressure on the system.
If the problem is that the contact point is in a depression, you can lay a pair of long boards across the depression to make it more level while pulling out. If the problem is that the contact point is a high point, you are pretty much stuck with what you are doing now.
Another route may be to add air bags to the truck that you can inflate/deflate from the driver's seat. Inflate them when pulling out to raise the rear of the truck, then deflate them when out of the driveway for normal towing.
Bobbo and Lin
2017 F-150 XLT 4x4 SuperCab w/Max Tow Package 3.5l EcoBoost V6
2017 Airstream Flying Cloud 23FB