If she was adopted by a step parent (or in any way adopted), she would have likely been issued a supplemental birth certificate with her new name and parents. My experience in tracking down docs has been that birth certificates for persons born before about 1950 seem to all be stored with their respective state's vital statistics, and not with the individual ciry/county/parish. Have you checked with Louisiana's vital statistics to see how you can go about getting her birth certificate or similar document verifying her name and DOB?
Do you know when her SSN was first issued? If it was issued prior to her marriage, that could potentially help you track down her last name.
Do you know when her SSN was first issued? If it was issued prior to her marriage, that could potentially help you track down her last name.