I searched on the subscription site Roots Ireland for all of Ireland and there were no 1810 Catherine Doran baptismal records in Co. Galway. There were four 1810 records in other counties. Also did not find the marriage record nor did I find any Fury baptismal records with parents Thomas Fury and Catherine Doran.
There is a Sarah Fury baptismal record in 1830 in Tuam with father Thomas Fury but the mother is Catherine Naughten.
Where did you find the 1810 baptismal record you cite? You posted in Ballynakill (Derrybrien) parish in Co. Galway. Do you have any info that points to that parish?