Burns Ancestors

I am looking for any ancestors of William James Burns that lived in the Moore Civil Parish around 1820. William was born around 1817 and his half brother Patrick was born around 1820. They both emigrated to the US in the 1840's where William was married in 1845 in Rutherford County, Tennessee to Jana Margaret Cotton. They later moved to Humphrey's County near McEwen.

In the 1850 census from Rutherford County, both William and Patrick are listed as being from Moore County or possibly Moore City in Ireland. Attached is the 1850 census which also lists the other family members living with them at that time.

William is my great great grand father and I am traveling to Ireland in June and would like to possibly meet any ancestors that remain there. Any information like birth or baptismal records would be helpful.

Thank you.