Interesting information, some new to me, found thanks to a Meta post about a resource at Southern Campaigns Revolutionary War Pension Statements & Rosters. I find new information here on a regular basis, it’s an incredible free resource for Revolutionary War researchers.
https://revwarapps.org/b414.pdf
Signatures of South Carolina Officers
Joseph Adair, Jr.
This is the first hard evidence I have seen proving much conjecture and family tradition that Joseph Adair, Jr. did indeed serve as Deputy Commissary of the Little River Regiment. I believe that this was the regiment made up of many of the men from Duncan Creek Presbyterian, under Col. Levi Casey. Joseph Jr. did not seek a pension, by all accounts he was quite comfortable and didn’t request one.
Joseph Adair Jr, Deputy Commissary signature
2647
Gambel, James, Account Audited (File No. 2647) Of Claims Growing Out Of The American Revolution. Date: 1776 C. or later

Full Document at the South Carolina State Archives:
https://www.archivesindex.sc.gov/ArchivesImages/S108092/S108092005000422000/images/S108092005000422000.pdf
Joseph Adair, Sr.
Joseph Adair Sr, Commissary signatures
637
Bourland, William, Account Audited (File No. 637) Of Claims Growing Out Of The American Revolution. Date: 1776 C. or later

Full Document at the South Carolina State Archives:
https://www.archivesindex.sc.gov/ArchivesImages/S108092/S108092001100382000/images/S108092001100382000.pdf
3302
Hanna, William, Account Audited (File No. 3302) Of Claims Growing Out Of The American Revolution. Date: 1776 C. or later

Full Document at the South Carolina State Archives:
https://www.archivesindex.sc.gov/ArchivesImages/S108092/S108092006600134000/images/S108092006600134000.pdf
Yes!!