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Willaim, obviously first up you want to find the Parish (out of 120) for your Feeneys. There would be a lot of Johns, & James isn't rare. But McCashin [spelling variations in Irl= McCashen, McCasheon, McCashon,McCassin, McCassion or without the Mc prefix] was rare in Co.Galway. Assuming John didn't move around much(he's not in Irish Constabulary index: that only started c.1833 & early records mightn't be complete)would be easier to find McCashins. Vast majority of people I'd told married someone living less than 5 miles away. I'm not aware of the Tithe Applotment Books [c.1824]being available on CD (Galway Fam Hist Socs have some?). I've checked "HouseHolders Index" [surnames on tithe & Griffiths] by LDS for Feeney etc & can email you a list of Pa with Feeneys in tithe [tithe didn't cover 100% of households]. Index for Grifiths Valuation CD (c.1855) only lists for ALL Irl McCashen, McCashin, McCashion & Cashin etc [no MacCas_] Only 1 in Co.Galway = Robert Cashin,Barrack St Oughterard [NW of Galway Town].My search of Catholic records there c.1810-50 noted only 1 John married [1831 & not to Ann]& no baptisms relevant to you. In Catholic Pa.Moycullen [Galway side of Oughterard]there's a baptism "1814 Aug 24 at Gortiloghlin Peter Feeny Bart: John 3/4" which I take to mean bapt of John Feeny son of Peter F_ (who is son of Bartholomew - shown to differentiate this Peter from another of same name living there)who donated 3 shillings 4 pence to the church. The Pa Priest died of cholea in 1823, so no records 1820-37. Marriages restart Oct 1843 & have John marrying then but not to Ann & no more Johns marrying till after 1850.Only bapts noticed with John father = the 1 that married in 1843.
It was illegal to be a Catholic up to 1829 & civil BDM registration for Catholics didn't start until 1864 & the uprising in 1922 destroyed many records. Nevertheless there are some Catholic records prior to 1829, although if I was in your position I'd start looking for John-Ann marriage as there are more records available then.
Your family history mentions the Shannon which puts them in East Galway say around Ballinasloe or Meelick where there were small pockets of Feeney families. There were about half a dozen such pockets, but most Feeneys in East were around Ballygar. Ballygar Catholic records go back to 1804, but the only place I know where they are accessible from overseas is the East Galway Fam Hist Soc & they have virtually all Catholic Records for East in their database [Watch Pa Glinsk & Kilbegnet - done by Roscommon FH, but part covers Galway & there are Feeneys there]. For a reasonable fee they should look up all their info in Catholic records for all East for all your names, so I'd suggest this is the easiest [maybe not quick?] way to try & find the Pa. There web page is on the Galway site.
Please email me with any comments or queries & for further info when Pa found. Also, the names of John's & Peter's sons in order of birth should provide a good guess at their father's name (& no doubt their father had a son of the same name). Bye for now, Ron