First off I'm really proud of you for going for those half days. You're not the only who gets sucked into a cycle of anxiety that prevents you from attending classes. Those half days mean a lot.
You are no longer a minor and don't need your mom's permission to see a dr. And you want to see a psychiatrist (not the same as a psychologist, who don't have an md and can't prescribe meds) specifically, they deal specifically with mental health and won't dismiss you like your family dr did. You can probably set up an appointment with a therapist at the same time, or get a referral from them. The most promising treatment is to take both meds and therapy at the same time.
Do you have health insurance, and the insurance card? You will need that info when you set up the appintment. Can you ask your mom for money for something else and use that towards paying the copay? I would recommend not telling your parents anything at all. My parents don't believe in depression or modern medicine, so I found it was much less stressful to just not tell them anything. Otherwise they would have tried to talk me out of "wasting" money. My health is between only my drs and me, its none of their business.
When you set up your online account with the health insurance agency, you can make sure all the bills and claims are sent electronically. You can tell the receptionist at your drs office that you would like them to not send any mail, in order to not jepardize your health. If you can't do that and pay the copay yourself, you can tell them to send the bill in the mail, that way your mom won't know until after the first appointment is over, the second has been scheduled, and the bill arrives.
For me the hardest part was making the phone call. Its worth a try showing up in person, it will depend on their policy if they accept walk-ins or if they can at least schedule the appointment in person instead of over the phone. Don't wait until things get worse, there might be a long waitlist. If you're going to skip school anyways, on next monday or tomorrow do you think you can get to a psychiatrist's or therapist's office?
As for school, you need to go, but not because of your grades. Don't put pressure on yourself about grades or perfection. The real problem is that if you skip once you are validating your fear (a fear which we both know isn't rational) and that will make it stronger the next time. Each time you give in to the fear it will get stronger, so pushing through the suffering now will save you from a lot of future pain. How about this: each time you start to feel that fear 12 hrs before, you get on a bus (what is your transportation?) and get to the cosmotology school. It doesn't matter that maybe no one is there and the doors are locked, what matters is not letting the fear win. Then you can reward yourself with a favorite snack or something else to associate the school with positive memories.
You deserve a better life than the one you're making yourself live. You've shown determination, by not giving up on school despite the last two colleges, and you're fighting for yourself. It will be hard, but I believe that you have the strength to fight your fears and to heal from this.