Legal Question in Family Law in Georgia

Yes.

And by the way, the marriage certificate in Georgia does not say if a woman (or man) has changed his or her name. The ONLY place that is found is on the application.

If a person says on the application they are changing their name and do not, the marriage is valid, but the person may have problems later for failing to change their name (for example, it is a criminal offense not to change one's name on a driver's license).

Note that many people in Georgia marry and choose on the application not to change their name and that either a husband or wife, or both, can change it by marriage.