This Adam Dickey Esq. of Hollybrook Randalstown, Antrim - listed under Nobility, Gentry and Clergy in Slater’s 1846 Directory is the Grandfather of Adam James Dickey born 1834 that came to New Zealand.
He was part of the Dickey family at Randalstown that owned the Bleeching Mill there on the River Maine.
Adam Dickey, Esq. of Culleybackey I would not think was the same Adam Dickey Esq. of Randalstown which this link refers.
Who knows there may have been a link between these two Dickey families.
I note that there ia a Ballydonnellan in County Galway and County Mayo. Did John Dickie come from one of these Counties?