Welcome to Ireland Reaching Out!
I did searches on Roots Ireland and Find My Past and could not find records for Cornelius, Charles or Michael. Corey was not a common surname in the 19th century and from the output I received, it would appear that Co.Tyrone, Co. Armagh and possibly Co. Clare are the likely counties for your ancestors. The surname Dougherty or Doherty is much more common in the Ulster counties.
Have you considered autosomal DNA testing?
You may also want to post your ancestor stories on our XO Chronicles site and possibly someone might recognize the family.