I searched ancestry.com and found the original application for his headstone. Apparently, all of the information you have found was based on this application. It was submitted by Harmon Murray, probably a descendant, on Feb. 19, 1940. As often happens, descendants get lots of information wrong. In this case, there was no 36th Alabama Cavalry. Church Murray's record is found with the 8th Confederate Cavalry. There is only one card in his file, and that is from the surrender at Citronelle, AL. He is shown in Company F, 8th Confederate Cavalry commanded by Col. W. B. Wade. It indicates that he was a resident of Lowndes County, Mississippi, and that he received his parole at Columbus, Mississippi, May 18, 1865. There is no further information in the files. The headstone application indicates that a headstone was shipped to Harmon Murray in Leakesville, Mississippi, on August 14, 1940. If it is not in the Slay Cemetery, it has evidently been stolen, destroyed, or even perhaps sunken so that it is not visible.