Irene, unfortunately I haven’t anything definite on bapt of Bridget or marriage of Andrew, so I’d suggest trying to narrow down likely districts in Co Galway where they probably came from, then intensifying your search in these districts. Maybe something in US could narrow down the district, such as church register for Bridget’s marriage if it has place of her birth, or Andrew’s death cert.
As for Co Galway things, registration of births didn’t start until 1864 in Ireland & ditto for Catholic marriages (1845 for the established church). Censuses& lot of church records were destroyed in 1922 turmoil. Because of the Irish system for naming their children, there are usually many of the same name clustered in the same districts & none in other districts.
As for what I’ve looked at, Griffith’s Valuation c.1855 lists heads of households. I only noticed 1 Andrew Feeny/Feeney (can check this from CD index if you want). He was in the townland of Tully, civil Parish of Killeroran, in the Poor Law Union of Mountbellow, Barony of Killian renting 5½ & 8¼ acres from Denis H Kelly, who had all the land around there. Mary & Laurence Feeny lived in the same townland. Nearby in Ballygar in same Pa lived Batholomew, & in Slievemurry were Peter & John. I would expect some of your Andrew’s sons to have these names if he came from the Ballygar district unless he had very few sons. The Tithe Applotment Book (June 1834) for there shows the Representatives of the late Andrew Feeney & Peter Feeney occupying land in Tully, & Law. Feeney in Tullylifs (Tullyliss?).
The only Andrew I noticed from my searches is a marriage in Catholic Pa of Castlegar in 1835, which is too early for you, & I didn’t notice any other Andrews there. I didn’t search baptisms 1854-70 there, so missed a vital period for you, but this Pa seems unlikely because of the lack of Andrews. I haven’t done much searching for these names in East of Co Galway where Ballygar is, so this could explain the lack of Andrews in my searches. Catholic Church records for Ballygar go back to nearly 1800, & have been put into a database by East Galway Family History Society (have info page on the net & all Catholic records for East on database, which they will search for a smallish fee). Also Tim Daly (was email@example.com) is descendant of Andrew Feeney born 24 Dec 1839 around Ballygar & has been researching Feeneys from there for years & so may havesome searches for Andrews from around Ballygar.
I’m trying to build a database of Galway Feeney descendants to put them in contact with each other, so please keep in contact – obtain my email address by clicking on my name at the top of this msg – I suggest you contact me direct if you wish more explanation or ideas, Ron