This isn't legal advice, and you should speak to someone who knows the law (maybe at your school) about actual C&F issues, but this is just what I recommend for your personal well-being:
Your school should have some kind of health services program that includes mental health services. You can seek help for problems such as general anxiety and receive counseling that is often incredibly helpful and can guide you in the right direction. They'll also help you determine if your problem is serious enough to warrant medication.
C&F can be an issue, potentially, but first and foremost should be your ability to thrive as a person and as a law student. If you need counseling or medication, you should be on them. You can deal with the C&F problems and still pass the bar a lot more easily than you can deal with
failing out of law school because your anxiety is crushing you and keeping you from doing what you need to do. C&F will require you to disclose any diagnosis and treatment you have ongoing, but hopefully you should be able to make a case by then that you're a capable person because of your treatment and it shouldn't be a reason to deny you bar passage.
And that's assuming it's a serious enough problem to require a formal diagnosis/treatment. It could just be regarded as social anxiety they help you overcome, and you could be well enough to not need treatment anymore by the time C&F rolls around.
Take care of yourself and what will help you. Right now that should be getting the help you need for this anxiety problem. You can deal with C&F when you're better and less anxious and it's time to actually deal with it.
Lastly, I do second the recommendations on doing activities to help you overcome anxiety. ToastMasters is probably an excellent opportunity if you can find a local chapter, or a debate team, or a leadership role in a small student organization that requires some (but not too much) speaking from an authority position to other members. Counseling and medication are only tools to make the process easier; to get through anxiety you still have to confront it and teach yourself that the things you're anxious about are not as harmful as you think.