After several days of work, I have identified the different David Moores living in Fayette and Lamar Counties from 1860 to 1900. Your Moores are not there. As noted, David and Sarah Moore, both born in Tennessee about 1825, are found on the 1860 census of Calhoun County with children Martha, Texas and Stokes, ages 12, 10 and 7, respectively. 'Sallie' Moore appears on the 1870 census of Calhoun County as head of household, age 45. Children Martha, Texas and James [Stokes] are in the family, as well as another son named John, born about 1862.
Your David Moore must have died between 1861 and 1870. Based on other data from the 1907 veteran's census and pension applications, we can positively identify other David Moores with men living elsewhere in 1860. At least three men by that name lived in Fayette County in 1860. I find Stokes moore living in Robertson County on the 1900 census of Texas, born Feb. 1853.
Since the David Moore who enlisted in the 1st Alabama Regiment enrolled at Benton AL, it makes sense that someone in the family might confuse him with your David Moore. But it's Benton, Lowndes County, rather than Benton County.
I can answer your questions about David Moores from Fayette and Lamar.