Amy, my GEDMatch kit number is T780556. I ran our kits together and we are definitely related. It was very interesting: if you leave the minimum segment length at the default of 7 cm, we show no match, but if you lower the minimum to 3 cm, we have matching segments on 9 different chromosomes (one of them just below the 7 cm threshold), for a total match of 35.3 cm, which is fairly significant. In fact, you're the first person whom I've matched on that many chromosomes in that branch of my family (although most of them have not taken DNA tests).
There are a number of families in the area to which I am related (McNulty, McDonnell, McAndrew, Loftus, Gill, Gallagher, King, Curraby/Creaby, Durkin, and McNicholas, among others), and there have been many intermarriages among those families, so I suspect that we are related several times over, through various lines (thus the large number of chromosomes involved). I have family trees posted at FamilySearch (the most detailed one), FTDNA, Wikitree, GEDMatch, MyHeriatge, and GENi, in case you want to check them out, and I'd be happy to exchange other info.
In the meantime, here's the info I have for the two families I mentioned, and their children. I do not seem to have noted the parents' marriages down, but they should be in the register somewhere, given the dates involved. I see now that the name was given as just Genty, rather than McGenty, but it would be the same family, and parish priests were notoriously sloppy and inconsistent with things like that (the priest often decided what to write, and the parishioners may not have been able to speak English).
That was the period of time when people started dropping the "Mac" and "Ó" portion of their name, at least in public records. Everyone in the area would still have been Irish speaking at the time (many in the area still spoke Irish at home into the 20th century, like my grandparents), so they would have used the Irish name Mag Fhinneachta ("son of Finneacht") among themselves (the "fh" letter combination becomes silent in Irish,, which is how the "g" sound became so prominent in the anglicized version of the name). I checked MacLysaght's "Surnames of Ireland", and it appears that, while the surname originated in Donegal, it is now most common in northern Mayo, anglicized as both Ginty and Genty. There are other Ulster names which are now common in the area (like McNulty), as a result of migrations during the Plantation of Ulster (and earlier migrations, as with the McDonnell's, who came from Antrim in the Middle Ages, and ultimately from Scotland)) Some people are now re-assuming the "Mc" form of their name, or the Irish form, but that has happened more widely in other families.
Note that some other Genty's are mentioned as sponsors at some of these baptisms. The two wives are both named Mary Genty, so I suspect that they were cousins, because some of the sponsors for their children seem to be the same people.
Also, when I went through the parish registers, I noted down families with the surnames I was looking for, and I knew of no McGenty connection at the time, so there are probably other McGenty marriages or baptisms in the Killasser parish register which I did not note down. These are all drawn from Microfilm 04223/06 for Killasser, and the image numbers are (i) for the frst family: 52, 68, & 56; (ii) for the second family: 55, 82, 96, 108, 120, 131, & 142. I'm mentioning that, because sometimes related families got together to take recently born children to church together to have them baptised, especially if the church was not close by. In Killasser, there was a church in the tiny hamlet of Killasser, and later also a sub-church over the hill to the west, in the townland of Callow, on the road running alongside the Callow Lakes (that road runs through Attymass as well).
Date Child Parents Sponsors Residence
16 Jul 1859 James Bartholomew McNulty / Mary Genty Bartholomew McNulty / Bridget Genty Carrick
30 Sep 1860 Bridget? Bartholomew McNulty / Mary Genty John Genty / Mary McNulty Carrick
29 Mar 1862 Patrick Bartholomew McNulty / Mary Genty John McNulty / Mary Genty Carrick
Date Child Parents Sponsors Residence
3 Mar 1860 Patrick John McNulty / Mary Genty John Genty / Mary Callaghan Carrick
17 Sep 1865 Mary Ann John McNulty / Mary Genty Henry Hughes / Mary McNulty Carrick
4 Apr 1868 John John McNulty / Mary Genty Pat Hughes / Mary McNulty Carrick
23 Apr 1870 Michael John McNulty / Mary Genty James Genty / Anne Higgins Carrick
9 May 1872 Martin John McNulty / Mary Genty Martin McNulty / Margaret McNulty Carrick
23 Apr 1874 Catherine John McNulty / Mary Genty Michael Hughes / Bridget Hughes Carrick
4 Jun 1876 John John McNulty / Mary Genty Michael Vigand? / Bridget Durkan Carrick