You won't be able to get a New Mexico license until you take care of your one year suspension in California. Even though the prosecutor decided not to take action on the case itself, you still have a suspended license for the refusal as far as DMV is concerned. Now, there is a way to vacate that suspension, and you don't have to file an appeal with DMV to get that result.
What you are dealing with right now is a "soft" suspension from DMV. That soft suspension changes to a "hard" suspension once the DMV receives an abstract from the Court verifying that you have been convicted of a DUI.
Without that abstract, your suspension, although it is in existence, can be overturned by showing DMV that your case has not been filed in Court. Hence, it is called a soft suspension and not a hard suspension.
You need to call DMV in Sacramento and request a form that essentially requests that DMV unsuspend your license. Once they send that form, you need to go to the District Attorney or City Attorney (depending on who has jurisdiction) and have them sign the form essentially stating that they do not intend to file your DUI case in Court, and that it was turned down because of so and so.
Once DMV receives that paper, they will vacate your suspension, and you can then get a New Mexico license. Unfortunately, a 1650 waiver won't work in your case because you were actually a resident of California and had a license in California when you were arrested for DUI.
Hope this helps!