Can you shower at a pool or something? (I actually do that a lot, our shower sucks!) Or maybe the YWCA or there may even be a shower at the theatre (think dressing rooms).
It sounds like you have a stressful job, but you shouldn't give up because of your own negativity. If they fire you, then at least it's not your fault and you can move on, but don't beat yourself up.
People do have bad days from time to time, and sleeping in happens. Most employers will be forgiving, as long as you communicate the fact to them. I would still call them or show up for your next shift, and apologize. You'll need some sort of explaination of why you didn't call, the only thing I can think of is that you didn't have access to a phone because you were too sick, or that you didn't know the number.