I would suggest that you contact the local SS office to find out what you can do about getting a replacement card.
Parental rights must be taken away by the court or I believe that they can be given up if there is another adult willing to adopt that child. I would suggest that you wait on this for the 3 months until you become residents of NC before you do anything about this. Give some time also to see if the mother is just adjusting to the situation or if she is indeed not interested in keeping a relationship with her daughter.
Depending on whether you are just separated, already divorced and whether or not you have an agreement, you should consult an attorney on the custody issue. It sounds as though you have primary legal and physical custody of your child, but if you had an agreement stating differently, you may need to file for it.
I commend you on attempting to keep the mother informed on matters concerning the child, but it sounds as though she does not want to be a part of her life. Do you have a custody agreement, separation agreement? I suggest keeping a record of any major decisions and your attempts to contact the mother. Do you want child support?
The best thing to do right now, is to go to SS office, request a new card. Explain the situation to them…you may have to pay, but it’s better than not having a card. Enroll her in Kindergarten. Do whatever necessary to take care of her and wait. Keep records of all money spent, all attempts to contact the mother, all information that you can think of. Send her certified letters to her work, with requests for information or informing her of issues. Keep copies. Take care of yourself and your daughter first.