The man you seek is probably the J.T. Reaves who enlisted with the Mobile County Militia on 12 Sept. 1864 for beats 8, 9, 10. This man was 16. There are no soldiers named Jesse Thomas Reaves who enlisted in the Confederate Army. The closest may be T.J. Reaves, a member of Co. D, 15th Confederate Inf. This man was wounded at 2nd Manassas in 1862 and again in 1864. he attained the rank of Corporal. This man was 18 at the time of enlistment on 13 July 1861 and lived at Ft. Browder, Ala. The reference on the grave may be the Alabama Volunteer Guards, a militia unit.
As for the 5000 tropps that landed at Claiborne. I believe it is a myth. Federal General Lucas raided Claiborne and Perdue Hill. His unit met scant resistance at Little River, a more determined stand at Shomo Creek at the site of Mt. Pleasant which led to a rout of the 15th Confederate Cavalry and the home folks that made a stand. Most of the 15th Confederate Cavalry, particularly T.J. English's command and men of Amos' command were riding fresher horses than the Federals and managed to escape across the Alabama at Gainestown ferry (also known as Sizemore's ferry and Ware's ferry in earlier times) after a running battle across Fletcher's branch of Waller's Creek. The troops of Lucas led by Chrysler entered Claiborne and Perdue Hill where Chrysler's men found the flag of the Claiborne Guards (and returned it well after the war). There was no resistance. Eventually, the troops entered Monroeville where they camped at the Baptist cemetery and along the Camden road. Some of the Federal troops returned to the Stockton/Blakely area. The main force continued towards Greenville with scouting/raiding parties scouring the country side for food and grain for their animals. The numerous claims after the war attests to the pillaging done by Federal troops.