How large is your school? Smaller schools might be more flexible switching classes, although larger ones may have more sections and teachers.
Do you know what you were supposed to take? Depending on your grade level, most subjects should be fairly certain -- eg, you should already have history in there, although it might be regular rather than honors and you'd need to change those details.
Make a list of what you expected to be signed up for, plus one or two alternates if that's full. Don't bend on core classes; they should have plenty of room.
You also said:
Now, there's a different teacher on my schedule--same class, though--and I don't know ANYTHING about her or her methods. I sent her an email with some questions, but how should I practice self advocacy in this situation?
It might be my timid personality talking, or differences between individual schools, but this doesn't seem like the best idea. At some schools, upperclassmen can occasionally request a specific teacher or two, but otherwise I think that might give the wrong impression. Just because you don't know the teacher doesn't mean she's not a good teacher; furthermore, school administrators won't admit to hiring a bad teacher, so they wouldn't let you avoid it barring extraordinary conflict.
Lastly, some counselors are too overloaded to listen to what they see as students' complaints, so it might be better to get a parent to ask if they won't budge and you're in completely wrong classes. Your counselor might think you can just be dismissed because of your age, unfortunately. On the author hand, they might think that if your parent asks you're immature, so maybe you can't win. YMMV.
