Tony and Tom, thank you so much for the further detailed information. Am I correct in assuming that I'll have to wait until I can get to Ireland to do this further research into parishes? I'm thinking that the National Library in Dublin has the parish records and could be searched in one place?
This is the only Alice Morgan who came up in a search for all counties in Ireland during a 10-year time span, so I'm pretty sure this is my ancestor. The only other close result was an Alice "Morigan", born April 1815 in Cahir Parish, Co. Tipperary. Could be a transcription error, but the Alice Morgan born in 1819 is a closer match.
Would it be possible for me to contact a heritage center in each county, Tipperary and Waterford, and try to search that way. After paying for numerous individual searches, it's probably as cost-effective to travel to Ireland, which, God willing and the creek don't rise, I hope to do within the next year!
Thanks again,
Joyce