You mention a DNA test so I'm just curious what your results were. Message me privately if you like.
I am a member of the Hamilton Family DNA project run by Gordon Hamilton and I have Hamilton ancestors in precisely the area of Antrim you describe. While they must have originated in Lanarkshire I know nothing about how or when they came over.
I would guess however we are not related because I'm in Hamilton Group I and I'm unaware of any other Group I members with Irish links. Any response would be appreciated.