Birthing at a hospital without OBGYN

Is it possible to just...go into a hospital to give birth if you don’t have an OB? I have insurance and I do currently have an OBGYN but I’m not 100% happy with them and part of me would just rather leave and continue my pregnancy alone and unbothered. Since I have insurance and the hospital is open to the public, could I still just go in? Probably a silly question lol

There’s only one hospital in my town that all the local OBs use. My insurance is accepted there. My biggest thing is just not wanting to be bothered at my appointments, I don’t want cervical checks because it’s only for their benefit and it isn’t necessary unless you’re in active labor. I have horrible anxiety and the thought of having to turn them down continuously makes me want to avoid it all together. I believe in myself enough to advocate for my birth while I’m in labor because pain makes me speak up more. So it’s just more that I don’t want to be bothered every week to check my cervix and feel pressured into it because my anxiety.