I suspect you are correct on both counts, and that he received the wound requiring hospitalization at Kinston, as he was admitted to C. S. A. General Hospital, No. 3, "Greensborro," N. C., March 11, 1865, having been transferred from Kinston.

03 08 1865 The Battle of Kinston, or Wise's Forks, NC.

At the Battle of Franklin, the 34th escaped the severest part of the fighting, but at Nashville, the remainder of the unit was nearly decimated. With the wreck of the Army, the regiment passed into the Carolinas where it skirmished at Kinston and again at Bentonville. Ultimately consolidated with the 24th and 28th regiments, about 100 of the original 1,000 members of the regiment were surrendered at High Point, North Carolina, 26 April 1865.
[http://files.usgwarchives.net/al/tallapoosa/military/civilwar/rosters/mt98uthe34tha.txt]

Unfortunately there are no record of patients, extant, for Goldsboro Gen. Hospital, No. 3. There are however, at the National Archives in Record Group No 109, a record of patients at Greensboro Gen. Hospital No, 3 for the period Jan. 1-Mar. 20, 1865 (ch. VI, Vol. 291)