He attended the University of Oregon in Eugene from 1959–1962, and was an All-American member of the golf team.

Shaw won four PGA Tour events and had over two dozen top-10 finishes.

Shaw claimed throughout his career on the PGA Tour to be four years younger than the age in some record books.

[3] He was suspected by some of being older, notably by Frank Hannigan, who as Executive Director of the United States Golf Association paired him with the then-19-year-old Ben Crenshaw and 24-year-old Johnny Miller, both known as fast players, for the first two rounds of the 1971 U.S. Open for his apparent amusement.

As it turned out, Hannigan was right; in 1988, Shaw produced a birth certificate proving that he had been born on the same date in 1938, which made him eligible for the Senior PGA Tour (now the Champions Tour) starting with the 1989 season.