Welcome here to the Forum. It is a horrible situation to be in, needing treatment, but without the help of Medicare.
While I guess you are quite right in thinking you will have to contact local doctors to obtain prices, and probably ask for a long consultation, do you have to do this alone? Is there a member of the family that can help you make the calls?
If it is a medical center it would be a good idea to ask for any GP that specializes or has an interest in Mental Health problems. Maybe that way you will get the best available treatment and a more expensive alternative such as a psychiatrist may not be needed, or at leat not so often.
One thing you can try is to contact a migrant center, if there is one in your area, they may have suggestions at to who to go see.
When you do find a doctor it can be pretty hard to explain everything face to face. I've got round this by writing things down beforehand in point form -a half page or perhaps a little more. I wrote about how I felt, waht my life was like, what was wrong. Then you only need to answer questions, not explain from the start. I've found it a lot easier, and the doctor has appreciated it too, having a list to work from.
The other advantage is you can take you time, and get everything down right before you start.
There will always be a temptation to do nothing, it is not as overwhelming/frightening/worrying at the start, however putting things off rarely helps. That was a mistake I made, and it simply made matter worse. Worse becsue I was not being treated, and when I finally did go to the doctor then it was that much harder to treat.
I would like it if you came back and said if you thought these suggestions practical