Dadswell played two seasons of NCAA hockey for the Cornell University, leading the Big Red to the Eastern College Athletic Conference title in 1985–86. His performance led to his being signed by the Flames in 1986. Dadswell turned pro the following season, spending the bulk of the year in the American Hockey League with the Moncton Golden Flames. He earned a call up to the NHL late in the 1986–87 season where he played two games with the Flames. In 1987–88, he won the backup position behind Mike Vernon and appeared in 25 more games with the Flames. However, when the Flames acquired Rick Wamsley late in the season, Dadswell found himself out of the NHL. He bounced around the minor leagues until 1993, but never returned to the NHL and retired from professional hockey in 1993.
In 2000, Dadswell was inducted into the Cornell Athletic Hall of Fame.