South Carolina didn't start recording births until 1911 so that option is out.
I'd look for their marriage license. Both sets of parents might be on it.
If not, I'd look for their newspaper marriage annnouncement.
Her parents might be in her newspaper death notice.
If you think Alexander is her father, look for his probate records. Hopefully she is mentioned in them. Chester County Clerk of Probate Court has probate records starting in 1787.
The Camden Archives and Museum, http://cityofcamden.org/departments/archives
has books, microfilm, maps, etc.
The SC Dept Archives and History, www.archives.sc.gov
also has wills 1782-1855. Successful Searching!