There is little in the way of Parish records this early but
you can try:www.emeraldancestors.com
free to view, pay for records
and an Irish History Foundation with parish records for baptism and marriages. You only need name and year to search, a baptism record will have both mother and father's names and mother's maiden name.
The Port might be Galway? not Galaway?
By 1862 Griffith Valuation has only 1 William Flaherty in the Parish of Inishmacsaint and 3 x Henry in the Parish of Magheeracross (probably 1 person). There are 297 Murphy's. this is used as a census substitute
The Parish of Inishmacsaint is partially in the Union of
Ballyshannon or in the Union of Enniskillen. This means the Parish is within 12 miles of the named town.
The Tithe Applotments 1820-1839: a census substitutewww.genealogy.nationalarchives.ie
1834 John Flaherty living in the Parish of Inishmacsaint in the Townland of Carrickboy.
Also you might try a Heritage or Museum Society as they often have journals with gravestone inscriptions!