A.M.
If you have been seeing an OB during your pregnancy, he cannot drop you due to a change in insurance especially that late in your pregnancy. You most likely will not get a doctor to accept you that late in your pregnancy as well. You may just have to pick a hospital you want to deliver at and show up when the baby is due and have the Hospitalist (OB On-Call for the hospital) deliver you.
I worked at LF in OB and the nursing staff is great and it shouldn't be that big of a deal. But, seriously if you are an established patient somewhere, they can't drop you legally as a patient. And, if you get on public aid, they cannot refuse you if they have been seeing you throughout your pregnancy. I would talk to them and don't let them think they can't see you.