Here are two accounts of his service. The 16th North Carolina was not involved at Chickamauga

Madison J. Wells

Residence Buncombe County NC; a 25 year-old Farmer.
Enlisted on 5/7/1861 at Buncombe County, NC as a Private.
On 5/7/1861 he mustered into "F" Co. NC 16th Infantry
He deserted on 6/4/1863

Sources used by Historical Data Systems, Inc.:
-North Carolina Troops 1861-65, A Roster
(c) Historical Data Systems, Inc. @ www.civilwardata.co

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M. J. Wells, Private, Company F, 16th North Carolina Infantry, Volunteer, enlisted May 7, 1861 at Asheville for 12 months, age 25, recorded Present, Sick, August 31, 1861, recorded at the Jan/Feb, 1862 muster as Absent, On Sick Furlough, Present at the Jan/Feb, 1863 muster, Deserted August 4, 1863 at Fredericksburg [Near Orange C. H., Va.], no further records

Note, his Company F muster rolls exist through October, 1864, last near Petersburg, Va.

Compiled Service Records of Confederate Soldiers Who Served in Organizations from the State of North Carolina
NARA M270