You can try doing a web search for "mental health services YOURCOUNTYNAME county" and see what comes up. You can also usually call United Way at 211 and ask who people with no insurance in your city/county can see. Sometimes it will be a county health center, sometimes it will be a non-profit agency, and sometimes it will be a for-profit agency that contracts with the county to provide services for the uninsured.

A web search for "YOURSTATE department of health and human services mental health care" should also turn up something.

These providers may have also been listed in your insurer's list or they may not.

I hope you find something. I would think that your insurer is obligated to give you at least ONE name of a provider that is currently accepting new patients with your insurance. Otherwise, they're not exactly "providing", are they? Sounds horribly frustrating.