You migth want to try to find the court record noted below. From the date - 1813, it might be a younger or different Hezekiah Doane.
KY County Court Records, Vol. II
Records Abstracts, Harrison Co., KY Books A, B, D, and D 1794-1828
p37. B/347. Hezekiah Doan is charged by Polly Sneed of this county, single
woman, as being the father of her child; Dec. 1813.