Covey attended Maranatha High School in Pasadena, California, where he played for the school's baseball team. He pitched to a 1.30 earned run average and a.131 batting average against in his four years at Maranatha. As a senior, Covey had a 7–1 win–loss record, a 0.40 ERA, and 138 strikeouts in innings pitched. However, he lost that year, and could not regain the weight. The Milwaukee Brewers selected Covey in the first round, with the fourteenth overall selection, of the 2010 Major League Baseball draft. He indicated to the Brewers that he was likely to sign with them, but that he wanted a $2 million signing bonus. During his post-draft physical, Covey was diagnosed with Type I diabetes. He decided not to sign with the Brewers, as he needed time to adjust to having this condition. Covey's party was quoted as saying, "They knew they would have to do some special things for Dylan." He ended up turning down a $1.6 million signing bonus offer from Milwaukee. Covey enrolled at the University of San Diego, and he played college baseball for the San Diego Toreros baseball team. At USD, Covey was near his parents and doctors who he trusted to help him manage his condition. As Covey regained weight, he struggled as a freshman, pitching to a 1–3 record, a 7.60 ERA, and almost as many walks as strikeouts. He improved in his sophomore year, with a 6–3 record and 3.32 ERA, but with 43 walks and 50 strikeouts across 81 innings. In 2012, he played collegiate summer baseball with the Orleans Firebirds of the Cape Cod Baseball League. As a junior, Covey had a 5–4 record, and a 5.05 ERA.
On December 8, 2016, Covey was selected by the Chicago White Sox in the 2016 Rule 5 draft. Covey made the White Sox' Opening Day roster in 2017, and he made his major league debut on April 14. Aside from five rehab starts with the AZL White Sox and the Charlotte Knights, Covey spent half of 2017 with the White Sox, pitching to a 0-7 record and a 7.71 ERA in 18 games. Covey was outrighted to the minors and removed from the 40-man roster on February 18, 2018. He began the season with Charlotte. His contract was purchased on April 28 to start the second game of a doubleheader. He was optioned back to Charlotte the next day and recalled once again on May 19. Covey finished with a 5-14 record in 27 appearances. On July 28th, 2019, Covey was sent back to triple-A Charlotte after a disastrous outing wherein he did not retire one batter. He finished 1-7 with a 6.99 ERA. Covey was designated for assignment by the White Sox on January 14, 2020, and outrighted on January 21. On January 22, 2020, Covey rejected the outright to AAA Charlotte and became a free agent.
