I checked the official lands records for Scotland and Morton Castle definitely did come through the lineage of the Kings of Scotland.Roughly (may be mistakes so do not take as correct): King of Scotland Malc.II d. 1034. Dau. of Malc. II, Princess Beatrice m. Crinan who d. 1045, issue: King of Scotland, Duncan I (murdered by MacBeth 1040, issue: King of Scotland, Malcolm II d. 1093 m. St. Margaret d.1093 and four of their children who appeared on land papers then connected with families of Dunbar, Douglas and Crichton in Scotland: Edgar b. 1074 d. 1106, Alex I, b.1077, d.1124 David I, b.1080, d.1553, Matilda b.1079, d.1118 m. King Hy I of Eng.
Brother of above King Duncan I was Malcolm who m. Algitha, grandau. of King of England, Ethelred II, Malc. & Algitha had son Gos-Patrick, 1st Earl of Dunbar and in 1067 Earl of Northumberland. Christina Bruce married Patrick of Dunbar, 7th Earl of Dunbar and March.
May be some mistakes in the foregoing, but it would give you a rough idea where the name Edgar came from.
The Adare/Adairs who descended from the Fitzgeralds of Ireland and whom are stated to have come from Adare in Limerick appear to be an entirely separate lineage.However, they appeared to have married into the abovewhen William Adair married a Dunbar c.1500.Then again it may have been and Edgar-Adare who m. Janet Dunbar.
The only slight connection I could find with the Fitzgerald Adares to do with Morton Castle was that Thomas Adair bought a small part of land near it or part of the estate.
Going by William Adair's statement c.1599 to the Government official that he was from Kildare and called himself Adare, I would not discount that theAdair/Fitzgerald lineage is true.
However, I have considerable doubt that the lineage for the Adairs of Bellgrove is correct in that it appears to me that they have tacked the Fitzgerald part at the beginning then appear somewhat to list the lineage of the Edgar Adairs.Perhaps only their own direct descendants would be able to clarify this point through any ancient papers which may still be in their family chests.