Jeanie- here's what I've found so far-
The following William's served in Tenneessee units-
William Jackson- Co. "F", 38th TN (Looney's)Inf.
William-Co. "C", 39th(W.M.Bradford's)Infantry
William K- 6th Mounted Infantry, Union
William- 8th Infantry, Union
William 8th Cavalry, Union
Of these units, only Co. "H" of the 39th Inf. and Cos. "B,C,D,H" of the 8th Cav(US)were raised in Greene County.
A check of Davis's in these units showed only the following officers-
Co. "B", 39th Infantry-Lt. Andrew W. Davis
Co. "E", 8th TN Cav(US) William B. Davis- 1st Lt. later promoted to Major.
No perfect fit but something to work from:-)