After a little further investigation. It does not appear there was any distinct incidents of cowardice, like fleeing the battlefield, recorded for Gholson's brigade.
Seems any label of 'cowardice' may have been derived from ongoing conflicts between state militia leadership and the Confederate army chain of command. Apparently Ghoslon's brigade of state troops built up over time a deep-seated reputation for unreliability, lack of discipline and high desertion rates and this caused other Confederate commanders to have serious doubts about Ghoslon's men. Wirt Adams, in particular, viewed Ghoslon's state cavalry brigade as unreliable and apparently he unsuccessfully suggested that the brigade be dismounted and its members dispersed among other commands to fix a perceived lack of discipline.
If all this is true, then any tag of 'cowardice' was probably unfitting for Ghoslon's group. But this undeserved reputation held in others minds may have 'rubbed off' on members of the brigade, like young Pvte. Howell - hence, the reference to 'cowardice' of the unit in his letter.
Incidentally, Pvte. Howell's
Find-a-Grave listing appears to be at,
https://www.findagrave.com/memorial/70476493/william-d-howell .
It shows his date of death as on 28 July 1864 (age 17-18 years) and there is also a photo of him. From the pictured grave marker, it appears that he was mortally wounded in battle at Atlanta, GA.