If you are looking at John Harwood, you might want to talk to Adamsons relatives. 2 have been writing to Rootsweb.
One said "My gt gt grandmother was an Adamson, married to Joshua Moore. The Adamson family once ran the Post Office in Waringstown. It passed into the hands of the Harwood family (my mother's people), when Elizabeth Moore married William John Harwood. My gt aunts ( William John's sisters), Judy and Hettie, inherited the Post Office and ran it until the late 1960's."
The author's e-mail address is not listed but another person wrote about an Australian Adamsons going back to claim an inheritance. They talked and were related.
The 1901 Waringstown census lists William John Harwood COI 35 postmaster with wife Elizabeth son Thomas Rurie? 7, Agnes Anna 5, and John Arthur 1.
There is a Samuel Beattie COI 43 baker with wife Bessie 39, son Samuel 17, Ellen 5, John Harwood 4, Joseph Alford 3 Thomas Matthew 1 with 2 others in the house. Possibly Bessie is a Harwood, but the date is far off for your Elizabeth.
More reasonable is the Margaret Harwood wife of Valentine Harrison possibly born 1842 of Ballydougan who has children William Valentine and James Henry in the 1860's.