That's great news about your dad. Many of the insurance counselors can be reached by phone, btw, so he doesn't have to go down there if he's tight on time. Let us know if he runs into any problems.
If you qualify for Medicaid or Medicare, he can sign up at any time. If not, he has 60 days from the date your insurance was cancelled to sign up for new coverage. And the good news is that open enrollment starts October 1st, so even if he's missed the 60-day window, you won't have to wait long for coverage.
One more thing: My parents also got divorced when I was 4 years younger than you. They had a very positive divorce, but divorce leaves scars no matter how well it's done. My sister and I went into therapy when we were your age, but neither of us thought it helped. Years later, we went into therapy as adults and it made a HUGE difference.
What I'm trying to say is: it's totally okay if you don't get counseling now. It may not even help you at this point. But you will probably have some scars and a good therapist can totally erase them. I ultimately went to a great university, I have a great career, an amazing husband, and I'm really happy (despite some annoying health anxiety). So you're going to be okay. :-)