While I want to actually visit the doctor, I've found it incredibly difficult to find one, apparently all calls get routed to the billing department

Fair point - it can be really hard to get in with a psych. I'd talk to your PCP and see if they can get you a referral at least.

She actually made a referral, then I found out that department wasn't accepting new patients... It's been a frustrating process.

That's very true. Sometimes the work only comes once you start to get better (and then don't think you need the help). This advice could work if someone has noticed a loved one needs help and has to seek out help on another's behalf. It took me years to finally reach the step of starting the search which was super hard. I'm getting good results though.