When we dispatch a bird here we use a killing cone. It is the best and easiest way for us that we have found. We make sure to have a very sharp knife to not make the bird suffer. However we need to find a better place to do it. Last time we did a hen, I thought my neighbor was not home and would not be bothered by the sight. Turns out she was, and let me know about it. (Oops sorry neighbor).
I really want to set up a portable butchering station, with a sink and cutting table and small frame to hang the bird from to pluck it. Maybe set it up behind the shed and then bring it into the garage to store when not in use.
I agree a gun is essential, but here in the city I can't shoot everything. I need to invest in a pellet gun at least. In back of our house is a city park/green belt with a creek. It is beautiful and so nice to not have neighbors right behind us. But we also get the wildlife that hides along the creek banks.
Plus the stray dogs and the dogs whos owners don't keep them on the leash.
Some one mentioned doing a meat bird swap. That would be great!
We do the egg hatching through the OSU extention office each year and last year we got about 14 meat birds. A friend came over to help butcher and we sent half home with him.
Thanks @Kyzmette she will get a good home here.

I can come get her Saturday. (Late Christmas present!) Thanks!!!
We are not doing much here today. I have 3 gifts to finish by tonight, so it is movie and couch day for us today. Plus I am brainstorming about re arranging the garden and getting dead tree branches down. I want to re paint the chicken coop red with white trim this spring.