Oh mine as well, but my grandma was a strong and stubborn woman, and clung to her ways. :) Snakes, especially ones large enough to swallow those chickens, trembled at the thought of her and her hoe.
I've not done chickens, (although I did catch/hook them for grandma, but I have done pheasants that were merely wounded. While it doesn't (usually) remove the head, grasping them by the head and sort of snapping, causes their body weight to snap their neck, to the same effect. Then you stick 'em in the game pouch, until you can get somewhere to clean 'em properly. I did learn a real slick way to do that too. Via a surgeon who had operated on my mother in law and was later invited to go to Nebraska (from Texas) to hunt on my in-law's farm. A bit hard to describe, but it involves a couple of quick snips with some scissors, and everything inside comes loose and is easily removed.