Donegal County..William Dougerey/Dougrey/Dougray/Dowrie married to Agnes Cook

Searching birth records for the William (c.1786, d. 1857) and his wife Agnes Cook c.1789, d1853). Their first 2 children John and William were born in Donegal, Ireland, but by 1818 their third child was born in Renfrew, Renfrewshire, Scotland. I would like to verify birth records of these two ancestors. I don't know if birthplace of "Donegal" means the city of Donegal or the county of Donegal.

Researching genealogy for my husband's paternall side..finding ancestors that were born in Ireland is a new discovery. The male surname has been spelled many ways over the years. I tried searching message board for name and came up with no results. I also tried Cook with no results.

Recent ancestors are all protestant, but that doesn't mean ancestors in Ireland were protestant. Any suggestions of where to start? Are birth records available online for the years I am trying to find?.Help much appreciated

Nancy Barclay