I think I know exactly how you feel. When depressed, it is like climbing a mountain to first figure out who to call, then call them, then go to the appointment. It seems like so many hurdles, and you just want to sleep.
But you HAVE to do it. Maybe not all at once. First, today, find a piece of paper and a pen and write down those three steps. Then you can go back to bed. Tomorrow, tackle the first step. Go to your insurance’s website, and search for psychiatrists. Make a list of names, phone numbers, and addresses. Then check that task off on your piece of paper with the steps. Then go back to bed if you want to. Then, Friday, call them, one by one. Just start at the top of the list. Ask if they are taking new patients. If they are, schedule an appointment, and you are done for the day. If they are not, move on to the next one. Once you’ve scheduled an appointment, check that off your list of steps. Do not throw away your list of names, you may need it again. You can then go back to bed. Now, your only job is to get to that appointment. I would arrange for someone to drive you there. That puts one more safety net in you getting up and going to the appointment. Put it in your calendar. Set alarms. Just get to that appointment.
You can do this. One step at a time, one foot in front of the other.