How do I find out whether I still have health insurance, and am I eligible for medicare?

Hello!

I've graduated from college a few days ago. For last few years I was on a college health insurance plan. And I should still have this plan, since this is paid till July 31.

The problem is- I'm not a student anymore. I don't know whether I still can use my insurance. Should it still work? How do I find out?

If I cannot use it can I get a refund for remaining two months?

And if I am not eligible for it anymore, can I enroll in Medicare or marketplace plan? It looks like there is a requirement that I shall not have any other insurance plan to enroll. But technically I have an insurance, which is unclear whether it is going to cover anything. Do I just enroll in marketplace plan, or do I need to terminate the student insurance myself?

Is there a direct answer to my question, or should I contact anyone? Should I call my insurance company, or the college?