OBGYN in network, but hospital was not

I will try to make this as quick and to the point as possible. My wife and I recently had our 3rd child. She had been seeing a particular OBGYN in our area that was in network on our health insurance plan. Well, when it came time to choose where to deliver we were told by the doctor “We deliver at hospital’s A and B”. We chose the one closer to us thinking everything was all good with the insurance. Fast forward 2 months later and come to find out, the hospital she delivered at was out of our network and we are stuck with a $14K bill. Insurance seemed to have covered things up to the OON allowances, etc. But $0 for my newborn and about $8k for my wife remains. I should mention we have Blue Cross Blue Shield

My questions are, shouldn’t her OBGYN have been familiar with our insurance when setting the hospital delivery? Where do I go from here? Am I totally screwed?

I’m trying to figure out if we have any recourse, or if we are sunk and I need to try and file for financial aid and bite the bullet.

Thanks