This may be a challange. It looks like there are at least 2,685 Willaim Davis listings, on both sides. Many William and W. Davis listings from Al. If we are lucky one of these may be the one. William C. Davis, pvt., Co. D 1st
Al. Cav. or W. C. Davis, pvt. co I 8th Al. Infan.

You need to take a bit of time and visit one of the major libraries in Al. and look through compiled services records or visit the Al. Dept of Archives and Records and look through the compiled service records and pension applications.

Good luck.

Wayne Adams