There's a small footnote in the service record for your Pvt Thomas J. Cash, 56th Georgia. As you mentioned, one of his cards states that his name appears on a "List of sick and wounded Confederate Prisoners delivered to the Confederate authorities at Mobile, Ala.". So, in looking up this particular record, the archivist fortunately left an additional clue at the bottom of that card: "Indorsement shows: 'Rec'd 347 of the above named men. – R.C. Forsyth, Lt. Col., P.A.C.S.'" This passenger count is that clue.
This very same "indorsement" and, in fact this otherwise identical record also shows up in the file of Pvt C. Stevens, of the 37th Alabama ... another one of those 347 men arriving at Mobile ... BUT, in Pvt. Steven's case, there is another record in his file that actually points to the name of the vessel transporting him and his fellow sick and wounded companions to Mobile ...
Steven's name appears on a "List of Surgeons, Ass't Surgeons, Attendants, and sick Confederate Prisoners of war captured in the capitulation of Vicksburg, Miss., July 4, 1863; now on board the U.S. steamer H. Chouteau en route for Mobile, Ala., via New Orleans, La. List dated Hdqtrs. Vicksburg, Grant's Army –."
Although Cash's file does not contain this record, by association, it seems most likely that he shared the journey aboard the same ship as a fellow traveler with Stevens and 345 other sick and wounded men and their attendants.
-Chip