You haven’t said what denomination Elizabeth was but in this case it probably doesn’t matter. The only likely record of someone in Ireland in the 1700s is usually from church baptism records but none of the various denominations in Ardstraw has any records for the 1700s at all, so I doubt you will find any information on her. The majority of migrants from Ireland in the 1700s were Protestant, mostly Presbyterians and some Church of Ireland. So she may have been of one of those denominations.
If Elizabeth’s family were farmers, and if they were still in the parish in 1834 then they would normally be listed in the tithe applotment records (which were a tax on land, so you had to have land to be listed). There were 7 Jack families there at that time. Whether any is related to Elizabeth I have no way of knowing but it does at least give you some idea of where the surname was found in Ardstraw at that date.
Research in Ireland in the 1700s is notoriously difficult due to the general lack of records. It would be very rare to find a record of someone born in 1727.