Diane:
Welcome to Ireland Reaching Out!
I read your paper with interest. I assume from your years of research that you are aware that many, many RC parishes do not have records back to 1810 or earlier. In fact, the baptismal records for Clonfert parish in Galway start in 1884.
This site https://www.swilson.info/sdist.php is maintained by one of our volunteers and uses the data from the 1848-1864 Griffiths Valuation data to show you how prominent surnames were by each county. Molloy is a common surname but was rarely found in Co. Cork so if both Andrew and Elizabeth were from the same county than likely it was not Co. Cork. The counties that show a high number of Corcoran and Molloy records are Mayo, Galway, Offaly and Dublin. However, this is a very general indicator.
I searched on two sites that have baptismal records. One site is a free site maintained by the Irish government www.irishgenealogy.ie This site has records for Co. Kerry, half of Co. Cork and Dublin city. The other site, Roots Ireland, is a subscription site and has records for the remaining counties but not for all parishes in some counties (Clare and Wexford for example). In any event I searched for a baptismal record for Andrew Corcoran and did find an 1802 record in Kingstown parish( better known as Dun Laoghaire) in South Co. Dublin. The transcription is below but interestingly the NOTES show twin of Larry. Is this is a coincidence since you have identified a Lawrence Corcoran born 1806? What was the oldest sons name? There were about five Elizabeth Molloy baptismal records born right before 1810 inclduing a few in Co. Dublin.
Hopefully, others will comment on your post. I do have a few questions on the DNA results but will hold off for now.
Let me know what questions you have.. Your Corcorans may very well be from Clonfert, Galway but more evidence is needed.
Roger McDonnell
Name:Andy CorcoranDate of Birth:
Date of Baptism:21-Apr-1802Address:DunlearyParish/District:KINGSTOWNGender:MaleCountyCo. Dublin
Denomination:Roman Catholic
Father:Peter CorcoranMother:CattyOccupation:
Sponsor 1 /
Informant 1:John CallehanSponsor 2 /
Informant 2:Mary Rule
Notes:TWIN OF LARRY