Valerie, I think you're doing just fine.
I will say that the neck-breaking method requires practice. We were afraid to try it, but when one of our chickens was savaged by a dog and appeared to be dead, my husband tried it figuring it couldn't hurt a dead chicken, and it didn't work. That was traumatic for us and possibly for the chicken, which was either not dead, or else just started reflexively flapping.
Chopping also requires practice and we have found that a kitchen cleaver is more controllable than an axe.
So I think you are very sensible to consider other methods. If you do decide to chop, practice first, have someone else stretch the neck out a bit and then step away while you chop down hard.
Sometimes I think the folks who say "oh, just get an axe" either learned very young or have simply done a lot of them and may not remember how difficult it was the first time. Also, men have more arm strength than women and are generally not afraid to hit something HARD, whereas ladies are often taught to be more peaceful!
No insult intended to men or ladies who are already good at chopping stuff. I just know that when I first met my husband I had a tough time even swinging a hammer hard because no one had ever taught me how to hit things right. Even now my husband is the one that dispatches the chickens, but at this point I have confidence that if need be, I can, with minimum trauma to the chicken.
Anyway, too much philosophy but I think sometimes this is what underlies the chicken-killing questions. Hope it helps with your thought process.