Allan, There was an Alex Adair in the Exchequer Department c.1679, possible under Sir Robert Howard.Sir Robert Howard evidently put him in charge of managing financial affairs in Ireland of Nell Gwynne.Nell's son, I think, (Duke of Albans, DeVere Beauclerk ?) was of Ardglass Castle, co. Down, N. Ireland.
Was this Alexander Adair the son of Sir Robert Adair, and brother of William Adair who dc.1661 ? Was this Alexander Adair also of Drumore Castle, Wigtownshire, married to Miss Agnew ?
Sir Robert Howard seemed to have dealings with Godolphin who c.1662 was made ward of Sir Robert Adair's grandson, Sir Robert Adair II who was then only a baby. William Adair's widow was Anna Helena (Scott) who remarried brother of Earl of Dundonald (Cochran ?) and also a Mr. Murphy of Dublin.
"Robin" Adair's dau, who married Barrington may later have been of Ardglass Castle, co. Down, N. Ireland.
I am not sure if this Alexander Adair was in your lineage.