I commute 11 miles one way in my cycling clothes, than shower at work and hang my cycling stuuf on my bike and walls in my cubicle. Whe it is time to go home, I put everything back on and go. I am in the south and commute in 100 degree weather and sweet like a mofo. After an 8 hour day at work my clothes are dry and ready to go.

You could always just keep you short on and out regular shorts over them and swap out your jersey with a tshirt while in class, since it is only 1 hr and 20 min. That is really not that long of a time and you wouldn't have to deal with changing clothes all of the time.