The Unit History for the 115th NY is available at the New York Military Museum's website:
https://museum.dmna.ny.gov/unit-history/infantry-2/115th-infantry-regiment . The Unit Roster gives his service record in the 115th as: ADAMS, HENRY.-Age, 37 years. Enlisted, August 4, 1862, at Saratoga Spa, to serve 3 years; mustered in as corporal, Co F, August 12, 1862; captured in action, September 15, and paroled, September 16, 1862, at Harper's Ferry, Va.; promoted sergeant, no date; wounded in action, February 20, 1864, at Olustee, Fla.; discharged for wounds December, 1864.
I couldn't read the other entry, old eyes, but if you can decipher the unit, there is probably a unit history and roster on the museum site. I think I have it D'Epineuil Zouaves
https://museum.dmna.ny.gov/unit-history/infantry-1/53rd-infantry-regiment
ADAMS, HENRY.-Age 35 years. Enlisted September 24, 1861, at Saratoga, to serve 3 years; mustered in as private, Co. A, September 27, 1861; transferred to Co. G, Seventeenth Infantry, March 16, 1862.