Hi, and welcome to the forum.

Unfortunately I have no experience with New Zealand, other than having met many Kiwis during a lovely stay in Tahiti. But one thing I can tell you based on my experience elsewhere in the world is that teaching credentials often don't transfer between countries very well.

You may want (or have) to look into private schools (which are usually free to hire according to their own requirements), or to research what is required for a teaching credential in New Zealand. In some countries I'm familiar with, public school teaching is considered a "civil service" job and is not open to non-nationals except on a temporary or exchange basis.
Cheers,
Bev