I've also seen the transcribed version online but am wondering if the original is actually this name or was it a different surname that the minister used this strange spelling for in his records.
Adsay doesn't come up in LDS site (under Ireland), PRONI Will Extracts, 1911 census (under Ireland), index to civil registraion (again all Ireland), Griffith's Valuation, Emerald Ancestors index, Ancestry index (under searches or family trees). All of which makes me think the name could be wrong in the church records.