Hi, I looked in Catholic parish registers for Elizabeth O'Neil's birth and found the following;
Elizabeth Neil
5 July 1799
parish: Nobber
county: Meath
father: Pat Neil
mother: Betty
Elizabeth Neil
4 June 1798
parish: Kilskyre/Ballinlough
county: Meath
father: Daniel Neil
mother:Eleanor Gaffney
Elizabeth O'Neill
10 Dec 1796
parish: Armagh
county: Armagh
father: Constantine O'Neill
mother: Mary Murtagh
Elizabeth O'Neil
6 March 1810
parish: Collinstown
county: Meath
father: Pat O'Neil
mother: Catherine Corrigan
There were no records for county Monaghan at all. It is possible she was baptised in a parish church closest to home but not necessarily in county Monaghan. It may be that records are illegible or no longer exist. Also, when I was looking at childrens' baptisms, they took place in a Church of Ireland church which may mean the family were Protestant and would not appear in Catholic registers.
Baptisms for children of Thomas Shaw and Elizabeth;
William Shaw
born 5 Nov 1826
address: Tullyree
parish: Donagh
county: Monaghan
Church of Ireland-St Salvatore's
father: Thomas Shaw
mother: Eliza Shaw
Sarah Shaw
bap: 18 March 1829
James Shaw
born: 13 Feb 1831
Margaret Shaw
born: 1 July 1840
* all the info for Sarah, James & Margaret is exactly the same as it is for William.
I could not find the births of any other children. It may be worth a try to contact St Salvatore's Church directly as you may find the others that way. An aside- this is the church where Paul McCartney married Heather Mills-it is on the Castle Leslie estate.
I also could find no births to Thomas & Elizabeth in Scotland. In the 1861 & 1871, I did find a Thomas Shaw who would be about the right age for your Thomas. There is a death of a Thomas Shaw on 3 June 1877 who may be yours. You can find it on Scotlandspeople. I'm thinking Elizabeth may have died in Ireland and Thomas & children may have moved to Scotland after.
It appears Thomas may have been married about 3 times. A birth exists for a Mary Shaw bap 13 Dec 1823 with Thomas Shaw & Mary Grirson as parents in same church in same parish.He also appears to have married again in Scotland. Hope this helps, Kim