I have seen one Carey with Z253. Carey can be an anglicisation if MacFiachra, chiefs of Oga Beathra and a clan of the Ui Fiachra Aidhne but of different origins to the other Ui Fiachra clans from Galway. The Og Beathra originated from Mallow, Co Cork, which would be consistent with having Z253. Anyway, feel free to join the Greater Galway DNA Project if you tested through FTDNA.