Alexy attended Twin Valley High School in Elverson, Pennsylvania. He committed to play college baseball at Radford University. During his senior season at Twin Valley, Alexy threw a complete game shutout in which he threw 164 pitches, leading his team to a 2-1 win. Alexy was drafted by the Los Angeles Dodgers in the 11th round of the 2016 MLB Draft, and signed with them for a $600,000 signing bonus, forgoing his commitment to Radford.
After signing, Alexy was assigned to the AZL Dodgers to make his professional debut; in 13.2 innings pitched for them, he posted a 1-0 record and a 4.61 ERA. He began the 2017 season with the Great Lakes Loons.
On July 31, 2017, the Dodgers traded Alexy, along with Willie Calhoun and Brendon Davis, to the Texas Rangers for Yu Darvish. The Rangers assigned him to the Hickory Crawdads, where he finished the season. In 19 starts for Great Lakes he was 2-6 with a 3.97 ERA, striking out 86 in 73.2 innings, and in five starts for Hickory he was 1-1 with a 3.05 ERA. He began 2018 with Hickory.