Your ancestor Thomas Thompson Watlington is NOT listed in the 5th VA Infantry book in the Howard series, but he is in the 5th VA Cavalry book, same series, by Robert J. Driver, Jr.
The entry in the book shows Watlington was born c.1825 and enlisted in Company C at Danville on 1 March 1862. Unfortunately, his military record ends 1 May 1862.
Hopefully, others my be able to provide additional information.
John P. Mann IV