Appeals and statutes of limitation are two entirely different things.

When it comes to products liability actions (which an action against a drug manufacturer would be) North Carolina not only has statutes of limitation but statutes of repose. A statute of repose provides an absolute cut off date beyond which you cannot sue, regardless of when you found out about the drug causing birth defects.

NC had a 6-year statute of repose, but the legislature changed this in 2009. The statute of repose is now 12 years. See NC GS § 1‑46.1:

Within 12 years an action –

(1) No action for the recovery of damages for personal injury, death, or damage to property based upon or arising out of any alleged defect or any failure in relation to a product shall be brought more than 12 years after the date of initial purchase for use or consumption.

An appeal implies that there was some decision in the case. An appeal has to be filed within 30 days after the court's ruling dismissing the case. I don't know when the decision was made in your case or if it was before or after the law changed. The change law in law usually only applies prospectively, meaning to cases filed on or after the date that the law took affect. If your case was filed prior to the new law taking effect, then there is nothing for you to appeal. I think this would be the situation even if your case was just decided as the old law would still apply.

However, I am not a products liability/personal injury attorney and you should discuss this with your attorney. If you do not have an attorney, then you need to have a consult with one.

Even though you may not be able to sue in North Carolina, there are many class actions and the laws may be different in other states and provide a longer period of time in which to sue. What I would do is try to find someone local, but if you cannot, then Google the name of the drug that allegedly caused the birth defects along with the words "lawsuit" or "class action" and see if you can have a free consult with an attorney who may be handling such cases.