Mark, here are some links I found for obtaining birth records in New Mexico. The catch is that you have to prove you are closely related to the person you are requesting information for, and I'm not sure if you have enough information to do that.
I think you may have to find out where the adoption of your father took place and see if you can get those records. Otherwise, I'm not sure whether you have enough evidence to request the records from New Mexico. But, there might be enough information here for you to call and find out whether it is really as complicated as it sounds:
Request in Person
NOTE: New Mexico birth certificates are restricted access records. State law restricts access to the registrant's immediate family members or those who represent tangible proof of legal interest in the requested record. Immediate Family means any of the following: mother, father, sibling, child, grandchild, current spouse, or maternal or paternal grandparent. Paternal grandparent is eligible if father is listed on the vital record. See the List of Acceptable Documents for Obtaining a Birth Record for details.