I’m impressed that you are trying to take charge of yourself. Good for you! Please don’t scold yourself for your homesickness. It sounds like maybe you didn’t get a lot of practice being away from home when you were little. Sleepovers and vacations at relatives’ houses are ways that some kids get used to being away from family. And please don’t scold your family either. Your folks are so concerned about your education, they’ve been willing to take on the challenges of home schooling. They’ve done the best they know how to get you out in the world — including supporting you in being a cheerleader.
The question is “now what?” I suggest that you talk to the doctor who is prescribing your medications about a referral to a cognitive behavioral therapist. Therapists with that training have a practical approach to anxiety and depression that I think will help you. If you want to research the approach a little, you could pick up a copy of the book Feeling Good by Dr. David D. Burns.
It’s unlikely that you can get rid of your anxiety in the next few weeks but with some effort, you should be able to get more comfortable with being away from home by the time the next tournament comes around.
I wish you well.