I would never carry a deer over my shoulder. In my day pack is a plasticized 6x8 tarp that is part of my "survival gear. I lay the tarp on the ground, roll the deer on it, and then gather up one end and tie the head into it. Then I measure out several feet of parachute cord and tie that into a loop at the head end of the tarp. I step into the loop and put the front of the loop behind my head and the two running lines go under my arm pits. And I start walking. Both hands are free and the deer and tarp just tags along.
You can make it easier by cutting a short, stout limb and rigging the chest rope like a water ski tow rope. Pushing on a stick is FAR easier than having a rope cutting into you.
