Tie the rope to one end, and then spin the branch until all the slack is coiled around it. Place the stick over one shoulder, squat down, spin up more slack, and stand. You want to lift as much of the deer's body off the ground as possible, with the stick providing leverage, much like a derrick. Now simply lean forward, and the leverage the stick provides will start the deer scooting along. Take slow, regular steps, and try not to waste energy by staggering back and forth.
Some hunters prefer to straddle the rope and use the stick as sort of a waist-high yoke.
If your back isn't as strong as you'd like, you can transfer all the work to your legs using this technique. Plod up the trail, and when you get to your gun and gear, stop and take a rest. Walk them another 50 yards up the trail, stroll back, and cool down. When you're ready, start dragging again. But getting there is the important part.