I suppose you have looked up articles in the Belfast Newspaper index for Jesse Adair. One which took my eye was his reward for bravery at Charlestown 4 Aug. 1775, Document ID bi, 106628 in the Belfast Newspaper index at:
In the book Topographical Dictionary of Ireland c.1850 it gives Charlestown, a parish in Barony of Ardee, co. Louth, Ireland, 2 miles north of Ardee, and five of the parishes constitute the Union of Philipstown.
The Belfast Newspaper gave his appointment on 23rd December 1770 at Plymouth.
His address appeared to be of Belfast.
Wonder if he was related to Thomas Benjamin Adair of Green Park and Loughanmore. Noticed that the Adair/Hodsons were of Green Park, Mullingar.