The Francis M. Lawson who's grave you found probably belonged to the following unit-

The Tennessee 1st Cavalry Regiment(CS) was organized January 7, 1862; reorganized May 24, 1862 with eight companies; reduced to battalion August 12, 1862, known as Battalion, but officially recognized as 13th Battalion; raised to regiment by addition of two more companies on December 16, 1862, and designated as 5th (McKenzie's) Cavalry Regiment. It was also known as the 1st (Rogers') East Tennessee Cavalry Regiment, 2nd (MeLin's) Tennessee Cavalry Regiment, 2nd Tennessee Cavalry Battalion, and the 13th Tennessee Cavalry Battalion.

There was also a Francis M. Lawson in the Federal 10th Tenn. Cav (US)and a Francis Lawson in the Confederate 22nd Tenn.Cav