What do I do if I want to become a US citizen but I still have my husband’s last name whom I’ve been separated for 10 years already?

Generally, a U.S. citizen who otherwise is eligible for naturalization, can have her legal name restored to her as part of the naturalization process, and this is true regardless of whether she has not obtained a divorce from a husband with whom she is separated. Alternatively, a Permanent Resident may apply for a legal name change (in Georgia, that is done through an action filed in the Superior Court where the person resides). There is no additional fee for restoring an applicant's original name (as shown on a birth record) in the naturalization application process, although for that the naturalization ceremony must take place in a court rather than at the USCIS naturalization auditorium. That contrasts with costs for filing fees and for placing an announcement in the newspaper when legally changing one's name through a Superior Court in Georgia. It would be wise for you to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex naturalization application process.