You are going through a difficult time right now. It is important that you have someone who can give you, not only the legal direction you need but also the emotional support. Make sure you take care of those things so it will have the least impact on you and your children. That is one of the things we stress with our clients in these types of situations. Having said that, you file a petition to establish paternity, legal decision making, parenting time and child support. The only tricky issue is how you go about serving her with the papers. I suggest you hire a lawyer to help you with your case.
You are absolutely entitled to child support. In fact,under Arizona law, you can seek child support for up to three years retroactively. But, you should file soon, because every month you wait now is a month you lose. The fact that she is in New Mexico should not be a problem at all.
A qualified and experienced family law attorney can of course assist you in getting where you need to be. Most of us offer free, 1/2 hour consultations, in which your matter can be discussed in detail. That would be a great opportunity to discuss any further procedural or substantive matters.