Hello Becky, at the time you are researching we rely on church records in the main, at that remove they are very patchy so it assists if you know the religion so we can search on what records there are remaining, even if any we cannot be sure if we have the correct one unless we know the parents.
I have looked at a site here https://jbhall.clahs.ie/ for County Louth and see a lot of Carr entries for grave inscriptions mainly in Drogheda with four mentions in Ballymacscanlan, (near Bellurga) it says 1779, here lyth the John Carr, also Magy Carr age 16, also Archibald Carr age 5 and Maria Carr age 3.
There is a site for county Down which covers Newry but not sure if it covers the piece in Armagh but a lot of Carr people listed but none jump out as yours, there is a list of Carrs so click on Carr when you find it, https://rosdavies.com/
You can record your family under the parish here and hopefully someone will be searching, you may also wish to try a few privage Facebook Groups such as Louth Genealogy, possibly Dundalk North End but this is mainly photos etc and there are some Carlingford and Cooley ones but again mainly dealing with photos and local things. There is a Carlingford website which went behind a paywall for photos some time ago, €15 annually I think, a lot of old photos on it. https://rosdavies.com/ I see there are Traynors but no Carr names.
Roots Ireland a subscription site has about 4 baptisms in Armagh in the 1870s father John and mother Deborah, religion is Lurgan Quakers. There is a Thomas Carr baptised to a James Carr and Rebecca Carr on 27th June 1802 which appears a bit late, in Armagh parish and Church of Ireland, there is a catholic family have children in Upper Creggan in Armagh also, no Thomas, this is near Newry and Dundalk.. Jphn Carr and Brodget Murphy are the parents, records here commence in 1796 but it is a bit far from Newry to be the correct one, RC parish registers are here but you can only search by month, year and event and can be a difficult read. The library in Dundalk is the research library but we all work off primarily the same records.
good luck
Pat