I see in ship lists that a William & Jane Adray came to the U.S. from Ireland in 1821 on the ship "Fair-Trader." He was 35 (born about 1786) and she was 30 (born about 1791). Was this your Wm. & Jane? [Source: NARA Supplemental Index to Passenger Lists of Vessels Arriving at Atlantic and Gulf Coast Ports (Excluding New York), 1820-1874. NARA publication #M334.]
You must have "Castlecaulfield, Donaghmore parish" as Wm. Ardrey's birth place from other sources. Maybe a bible record, obituary or tombstone? It would help to know what source gave that specific location.
Several on-line queries and messages have William Ardrey born Sept. 7, 1792 Ireland, died Oct. 26, 1869 Lewiston Niagara County, New York. Findagrave has a burial record, but no photo of his tombstone:
William Ardrey Birth: Sep. 7, 1792, Ireland Death: Oct. 26, 1869 Lewiston, Niagara County New York, USA
Family links: Children: Jane Ardrey Lightfoot (1816 - 1907)* *Calculated relationship
Burial: First Presbyterian Church Cemetery Lewiston, Niagara County, NY ------------------------------------------------------------
Back to records in Ireland, Castlecaulfield has a wonderful Church of Ireland, St. Michael's, with very old records dating back to 1748.
Most of the Donaghmore parish Church of Ireland records (St. Michael's) have been microfilmed and are available through your local LDS branch library. See their FHL catalog on https://familysearch.org and do a place name search for Donaghmore, Tyrone, Ireland, to see what the film numbers are for available church records. Some films have copies of the original handwritten church ledgers, and other films have typed abstracts.
The most extensive collection of Co. Tyrone church records, including Donaghmore parish, is at PRONI in Belfast.
For Ardreys in various County Tyrone records, including church records, tithes, marriages and other records, see www.cotyroneireland.com
I'm still searching for a Donaghmore/Pomeroy parish bride of unknown "A-- " surname who was born about 1773 at Thornhill (less than two miles directly north of the town of Donaghmore). Mary Ann A-- married David J. Armstrong in 1806 and emigrated to Quebec with him and 6 children in 1817. If she turns out to be an Ardrey, I'll be glad to share anything else I find on her.