No stranger to championships, Oxley paved the way for Rayburn's first district football title in the late 1960s before embarking on a collegiate career where he contributed to three consecutive Southwest Conference championships for the Darrell Royal-coached Texas Longhorns. 

As a senior at Rayburn High School, Oxley earned first-team All-District honors as both an offensive and defensive lineman, helping the Texans secure their first-ever district title. During a crucial district showdown, Rayburn defeated Pasadena 34-7. After the season, he was recognized with All-State honors on both sides of the ball and played in the Texas High School Coaches Association All-Star Game. He also lettered in basketball during his time at Rayburn. 

At the University of Texas, Oxley lettered three seasons as a Longhorn offensive lineman. He played a role in UT's national championship run in 1970 as a sophomore. By 1972, he had become one of the team's top talents, breaking into the starting lineup alongside All-American Jerry Sisemore and blocking for All-American Roosevelt Leaks. Following a loss to Oklahoma, Oxley was moved from guard to tackle during a line adjustment, after which UT won its next six games and then triumphed over Alabama in the Cotton Bowl. He was named second-team All-SWC after the season. Notably, Oxley was also an outstanding student and was named a Football Academic All-American by the College Sports Information Directors of America.