It appears that the Pvt William A. Norman that served in the 8th Cavalry Regiment and Harris's Battery was from Scott County. He served in a Scott County company in the Missouri State Guard prior to enlisting in Confederate service. Here is the record I have for him:

Willliam A. Norman: Age: 28. Served as a private in the 4th Infantry Regiment (Waugh's), 1st Division, MO State Guard. Enlisted as a Pvt in Co H, 8th MO Cavalry Regiment on 5 Oct 1862 in Scott County, Missouri. Transferred at date not of record to the 4th MO Field Battery (Harris's). Captured at Mine Creek, KS on 25 Oct 1864. Transferred to Rock Island, Ill Military Prison and died there of disease on 8 Feb 1865. He is buried in Grave #540 in the Confederate Cemetery at that locale.