Prior to 1845 there are very few records for Roman Catholics. The place to start is the Griffiths Valuations. Drumragh Parish is the most likely you are after, and there is a James Woods of Gillygooley, and a William Woods of Gortmore there as land owners in 1851 when the Griffiths Valuation was done in County Tyrone.
Termonmaguirk (East and West) also is now Omagh.
I will keep an eye out for you, as I am also looking at WOODS in Co-Tyrone for siblings of Rachel Woods B:circa 1792, Married: 1813 to Patrick McCallion (B:1793) from Fintona, Parish Donacavey.
A suggestion of the mothers surname possibly Taaffe. That is the most likely to appear similar in the old handwriting of the day
