Yup, I have the same insurance and the same problem. Found that out with my first daughter. And check this out. My second daughter was born in Sept, we reached our max prev care by Nov. With one set of shots and couple well visits. Its crazy isn't it. My amt is $500. Yours may be the same or I know sometimes its only $250.
But to answer your question. Yes, you have to go to the health dept as far as I know. I do not know of any CVS or any place like that, that does them. However, that being said, my pediatrician's office does give the county shots after the amt has been met for just the admin charge, just like the health dept. So yes, some pediatricians do that. It is nice, because waiting at the health dept can take a couple hours. The first year is the tough one, but usually after the first year, you are fine. My oldest is 2 1/2 and we never meet hers now.
I dislike it very much too, but I have to admit besides this one problem, my insurance is wonderful! It even covered my IUD. So I can't complain.