I think what I'd do is call this woman and say something like "Hey, I'm going to sign up for that class tomorrow and I just remembered that you asked me to sign your daughter up too. I know you mentioned you hoped I could drive but I'm not going to be able to do that. Did you still want me to sign her up?"
If the mom says yes, and there are fees involved you can say something like "oh ok, so should I tell them you'll be dropping off a check later this week?"
I don't think you have to give any explanation why you can't drive but if she asks you can just say that while you don't mind helping out once in a while if your friend has a conflict, you can't commit to doing it for each class. You don't need to explain yourself any further than that. But if that seems too blunt you can soften it a bit by throwing in there that you have a lot going on in the next few months. It makes it clear that the mom is responsible for her own kid, and that if she NEEDS you to drive, she should check with you first.
This way, her daughter doesn't miss out on the class. If you don't say anything and the mom assumes you've signed her up, the deadline might pass and the child might not be able to enroll. By saying something you assert yourself and hopefully nip any future issues in the bud while not punishing the child for her mom's very odd behavior.
Also, I think sometimes it's all in the tone. Since you said you like this woman, I'd just keep it easy breezy--very light. As long as you don't sound put out or irritated, it should go over fine.