Are there sports history classes at the university level?

It depends on the university, of course, but many do offer such classes. They may be offered through the history department, or sometimes as part of a sports-related major.

We had some interesting classes available at UC Santa Barbara in the Coaching Department. They offered a Minor degree, which included Sports Appreciation 1A, 1B, and 1C. Each quarter focused on the sports of that season, including Fly Fishing, La Crosse, Rugby, Archery, Curling, Biathlon, etc.

Each of the ten weeks of the quarter would focus on one sport, including the history of the sport, evolution of the rules, and how it is played.

Michigan State offers HST 324, sports history. It’s about halfway down the page. It’s an online course, so you could take it now.