Born in New York in 1915, Dawbarn and his brother W. Lennox Dawbarn were sons of Alice Carroll Dawborn, who in 1920 remarried, to Baltimore, Maryland lawyer and USF&G director Robert Dixon Bartlett.
He married Mary Lawton Dawbarn (1915 - 1967), with whom he had a son, Henry Dunlop Dawbarn (1942 - 2008).