This is so foreign to me as here we have publicly funded health care. While one can obtain private (user pay) therapy, psychiatric care is generally covered.

That being said, I have a few ideas for you...

See your family doctor. He/she may have a working relationship with some psychiatrists and might be able to make suggestions or referrals based on knowledge they already have of your situation and who might be the most likely to have a positive working relationship with you.

Seek out some advice from your local Mental Health Care association. They will have a list of those psychiatrists taking new patients and again may be able to point you to ones who may be most suitable.

Find out if your employer has an EAP (employee assistance plan). These are very common these days. They provide a free and confidential service that can assist you with just such your query. They can put you in touch with 2-3 psychiatrists that your insurance plan best cover. In most cases, the EAP will also cover the cost of consulting with these professionals.