Do you use the axe very much? You might start by practicing with the axe you already have, on things like sticks or cornstalks. They are not exactly like chicken necks, but they can provide good practice in hitting the spot you are aiming at.
If the axe is too heavy for you to comfortably use, you might consider something like a hatchet, machete, or cleaver. Yes, the PVC cutters also sound like a good thing to look into.
But I would definitely consider practicing with the tool you have, on things that are not a live chicken, before you put much effort into other ideas. Sometimes 5 minutes of practice can make it obvious whether to continue practicing or try something else.