You think wrong. NZ citizens can definitely apply for a NON O for Thailand, and I would recommend that this is by far the best option in this situation.
Sounds like you need to go back and read the conditions of your visa. It is always important to know exactly what visa you have, the conditions, and the requirements for extending it. If you are planning to get the 2nd 5 years, while inside Thailand, you better get organised very soon. But by the sounds of it, you were not anticipating the 2nd 5 year application, so head back home and get another 5 years.