I'm so sorry about this Katie. You'll be receiving all sorts of bills, even after your insurance has paid. Depending upon what type of insurance you have, you may or may not be responsible for these. But that doesn't keep them from billing you. For example, HMOs and PPOs negotiate much lower rates for services and the hospitals, etc. are required to accept the lower rates in order to be a part of that "network". Legally, you are not required to pay the difference between the billed amount and the amount paid by the insurance company in those cases, but many people don't realize that. And providers routinely bill the patient for the difference because often, the patient will pay it since they don't know they don't have to. A call to your insurance company after they have paid will give you more information. Of course, if it is out of network, different rules may apply. Just don't be too quick to pay anything until the dust settles.
I know that probably rubs you the wrong way since I suspect you are someone who pays bills promptly and isn't used to having a creditor call - and they will call! But just screen your calls for a while and know that this may take up to 6 months before it's all settled.
Hang in there. It's awful, I know.