As far as I can tell, there was no "Battle of Adams Run," although there were a couple of battles, Pocotaligo and Johns Island, in the area. However, Henry did not die in battle.
The file for payment of funds owing to his widow, Susan, contains the following:
"Died of typhoid fever at the hospital at Adams Run, South Carolina on "7 June 1862.""
Page 16 (Front)
Treasury Office, 2nd Auditor's Office,
March 26th 1864
The Adjutant and Inspector General, C.S.,
Please report me the term of enlistment and the service and death, &c., of Henry Trottle last of Captain Sharpes Company 1st Regiment S.C. Cavalry. Alleged, died 7th June 1862.
(signed)* Calvert, Ch. Div Deceased Soldiers."
In different handwriting:
"Enlisted Decr. 4th 1861 for the War
Died June the 2, 1862"
Back (Page 17)
"2nd Auditor's Office, Dec. 15th, 64
Henry Trotter Private Co F (Capt Sharp's) 1st S.C. Cavalry Regt.
Enlisted 4th Dec 1861 for War.
Last Paid 30th Apl. 1862. Commutation & Bounty still due, none having been paid on the Rolls in this Office. Died June 2, 1862."("South Carolina Civil War Service Records of Confederate Soldiers, 1861-1865," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:XGKZ-6P1 : accessed 11 June 2016), Henry Trotter, 1862; from "Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of South Carolina," database, Fold3.com (http://www.fold3.com : n.d.); citing military unit First Cavalry, NARA microfilm publication M267 (Washington, D.C.: National Archives and Records Administration, 1959), roll 7.)