Been in the same situation...

What you need to do is this: talk to a doctor, get a note deferring the final exam if that's what will help. Talk to your prof, explain the situation and ask for an extension on things...

If that fails, you can drop a class after the dropdate has passed, but I think it appears to be negative on your transcript. But it's an option you shouldn't consider until you've looked into the other two things.