Norm attended Fairfax High School in Los Angeles, and signed with the Dodgers while they were still in their original home of Brooklyn in . His brothers, George and Larry Sherry, were pitchers in professional baseball. Larry had a successful MLB career as a relief pitcher and was the Most Valuable Player of the 1959 World Series; he was Norm's teammate from 1959 through 1962.
A right-handed hitter, Norm Sherry spent seven years working his way through the Dodger farm system, and another two in military service. By the time Norm reached the Dodgers, in 1959 for a two-game "cup of coffee," he was 28 years of age and the team was based in Los Angeles.
Sherry made the team as second-string backstop (behind John Roseboro) from through . Early in that tenure, Sherry took aside the prodigal Koufax - who was struggling to become a consistent winner in the majors despite a blazing fastball and one of the best curveballs of all time - and convinced him to take something off his fastball to get better control. The results were astounding: despite finger and arm miseries, Koufax dominated the National League from 1962 through his swan song, winning three Cy Young Awards and leading the Dodgers to three NL championships and world titles in 1963 and 1965.
His average plummeted to .256 and then to .182 (1962).
The Dodgers sold his contract to the lowly New York Mets on October 11, 1962, to afford Sherry more playing time, but he batted only .136 in a career-high 63 games played (and 147 at-bats) in New York in , and his major league playing career ended.
All told, in 194 games over all or part of 5 seasons, Sherry batted .215 with 18 home runs, and .288 with runners in scoring position.
In 1965, Sherry began his managerial career in the Dodger organization, and switched to the California Angels system in .
Williams had been extremely successful in his previous terms with the Boston Red Sox and Oakland Athletics, but his cold and hard-edged demeanor did not go over well with a losing Angels club. The Halos were 18 games under the .500 mark on July 23, 1976, and in the midst of a player revolt when Williams was given his walking papers.
Sherry, named his replacement, salvaged the season somewhat with a 37-29 record as skipper. That winter, the Angels signed high profile free agents such as Bobby Grich and Joe Rudi and expected to contend in the American League West in . But the team struggled and was only 39-42 and in 5th place on July 11 when Sherry was released in favor of his third-base coach, Dave Garcia. The firing marked the end of his major league managing career, with a career ledger of 76 wins and 71 defeats (.517).
However, Sherry returned to the coaching ranks, ultimately as an "official" pitching coach, working with Williams with the Montreal Expos and San Diego Padres, and with another ex-Dodger, Roger Craig, with the San Francisco Giants.