My health insurance denied two of my claims after telling me they are covered under my plan through a recorded phone call. Can I file a small claims court case?

I have been going to the same family doctor and gyno for the last couple years. It’s always been covered. Early this year I called my insurance before a procedure with my obgyn to make sure it is covered under my plan and they said yes. I also go to family doctor for a yearly checkup. Months later, I receive a bill for the entire amount! I call back twice within the first two months of receiving the bill and both times I am told there was a mistake and the doctor and procedure should both be in network and they send the claim through again. Fast forward to now and they claim it is out of network so I appeal. While in the phone call to appeal, THE LADY SAYS THERE IS A MISTAKE AND IT SHOULD BE COVERED! I wait and now, the appeal has been rejected and I owe around $2,500. This has caused me so much stress. I am a college student and work full time to pay for my own tuition and living expenses. If I would have known I was going to owe this much, I would have never gone to the dr in the first place. I am thinking of going to small claims court but never done it before. What can I do? Are calls with insurance always recorded and stored? Can I use that as evidence?
Additionally, it rubs me the wrong way that my obgyn and doctor have been in network with my plan and suddenly they are not? Shouldn’t I be informed of this? I am so confused.