Fairfax Harrison's book (The Virginia Carys) has your gg grandmother's birth parents as Miles Cary and Ariana Digges, although no birth date is given.

Sources given are:
(1) Statement of Miss Susan Cary, 1868.
(2) Chancery papers, Cary v. Hill, 1817, William & Mary xx, 67
(3) Swem and Williams, Register, 1918.