Generally the answer is sort of. They take your insurance information, and are not allowed to lie about being in network. That's insurance fraud and they get in big trouble.

They usually make you sign something agreeing to pay whatever insurance won't. This is sort of redundant, but you essentially are confirming that you understand that insurance may not fully cover your care. You should not have to sign this at an in network facility when undergoing non-elective care. Exceptions arise for emergency care where you sign on a blanket basis, because it's likely to overrun your premiums.

They are not legally required to tell you what the price will be and they are not legally required to honor the price advertised when they bill you. Importantly, they also can't just lie. They're bound by what's called "good faith". As long as they price they give you is honestly their best guess, they're off the hook.

Forcing medical providers, especially hospitals to publish and honor the price of their services not subject to insurance negotiation was a big part of Trumps healthcare reform initiatives, along with right to try and other misc small items.