Swinson joined the WVU squad in 1973. As a freshman, he was listed the second tight end behind Hay Sheehan. WVU Coach Bowden said, "Randy is strong, rangy, and possesses good hands; he will be an outstanding player one of these days at WVU." In 1974, he was listed as the top following spring drills. He was an outstanding blocker, playing nine games and starting five in 1974. He caught 19 passes for 278 yards and one TD. He had a good spring practice in 1974 and was named most consistent receiver by the coaches. He caught six tosses for 62 yards in final Gold-Blue game. As a senior, Swinson caught six passes for 86 yards in the spring game in 1975. He was the third-leading receiver in 1975 with 19 catches for 282 yards. In 1976 Swinson caught three passes for 49 yards in the spring Blue-Gold Game as a tight end. Randy majored in physical education. He was All-Metro at Spingarn High School.