Davidson attended Baylor University, and played professionally for the National Football League's Baltimore Colts (1954, 1957), and the American Football League's Dallas Texans (1960–1962) and Oakland Raiders (1962–1968). Davidson also played quarterback for the Fort Bliss Falcons in 1955 to 1957. A game between the Fort Bliss Falcons and the Fort Sill, Oklahoma Cannoneers was played for a trophy called "The Little Brown Dud." The Cannoneers won the game and took home the Little Brown Dud. Cotton was awarded All-Army Quarterback in 1955.
Davidson was selected in the first round of the 1954 NFL Draft by the Baltimore Colts. In addition to playing quarterback, he also was a placekicker and punter. An original Dallas Texan, Davidson was the first starting quarterback for the franchise.
On July 2, 1962, the Texans signed Len Dawson, thus ending his career as the Dallas quarterback. After the 1962 season opener, he was traded to the Oakland Raiders for the first overall selection in the 1963 American Football League Draft, which was used by the Chiefs to select future Hall of FamerBuck Buchanan. Just two weeks later, Davidson started at quarterback in his first game with the Raiders. Playing against his former team, he threw for 248 yards and rushed for a touchdown in a 26-16 loss to the Texans.
The high moments of his career were being selected to the American Football League All-Star Game twice: in 1961 and in 1963. He was honored as the MVP of the 1961 AFL All-Star Game while with the Texans.
Davidson finished with an all-time record of 19-33-1 as a starter.