Howdy and welcome!
Are you sure there aren't showers and lockers at the gym? If not, is there anyplace you can keep some toiletries on campus? If not, you'll just have to carry them with you.
I teach at a community college with no showers on my campus. I keep my toiletries in the office I share with other faculty. I clean up in the restroom using Wet Ones. I put on fresh deodorant and spray with a lightly-scented body spray. It's extremely hot and humid here in Texas and I was riding six miles each way during the summer semester but I still felt fresh and clean after I cooled off.
Be sure to catch the newbie stickie at the top of this forum. Good luck and have fun!