I would not change OBs because of this situation. Many doctors are not fully aware of all the details of how their office staff does things. So, to me this problem doesn't so much seem to be a problem with the doctor's qualifications as it does with how the doctor manages their office (I agree this can be an important aspect in finding a healthcare provider, but you are really only committed to this one for a few more weeks).
You have alot of time already invested with this doctor and the most important thing is that you trust the doctor to deliver your baby. IMHO, if you were that concerned about your choice of hospital, it would have been far better to switch OBs at the point in time the doctor changed their affiliations.
Before you cut ties with the current OB, I'd also suggest you find out how much another doc would want upfront to take someone so close to their delivery date - you might find that to be more expensive than you might anticipate. Also, keep in mind that doctors are a tight bunch and if you mention the reason you're changing, you're not likely to start out on the best foot with the new doctor as it might seem (irregardless of the situation) that your only concern is getting a "good deal". So, I urge you to be cautious in approaching this situation and consider what's the best interest in terms of your baby having a healthy delivery.