Shough spent three seasons with the Ducks, helping the program capture the 2020 Pac-12 championship. He is expected to graduate this spring and will have three years of eligibility remaining.
He played in 11 games for Oregon, amassing 1,703 yards, 16 touchdowns, and six interceptions. Shough added 282 rushing yards and two scores on the ground.
The 21-year-old served as Justin Herbert's backup in 2019 before becoming a starter in 2020.
Shough now joins a prolific aerial offense that has led the Big 12 in passing yards in four of the last six campaigns. He'll likely compete with junior signal-caller Henry Colombi to replace former Red Raiders starting quarterback Alan Bowman, who entered the transfer portal last month.