ER visit while on vacation - got a $9,400 bill because the hospital was out of network. What are my options?

My daughter had an emergency appendectomy while we were visiting family in another state over spring break. She was in serious pain, we took her to the nearest emergency room, she had surgery, and thank god she's fine. We were there two nights total. I gave them my insurance card at intake. When I got home I assumed everything would be covered the same as usual - we have a $1,500 deductible and then 80/20 coverage. Seemed straightforward.

The explanation of benefits came in last week. The hospital is out of network. The surgeon is out of network. The anesthesiologist is out of network. Total billed: $41,200. My insurer processed it at their "usual and customary" rate, which they calculated as $22,800. After my out-of-pocket maximum of $6,500, they're saying I still owe $9,400 to the hospital because their actual charges exceed the UCR amount. I had no idea I could even have exposure above my out-of-pocket maximum.

A few questions for people who know this stuff: Does the No Surprises Act apply here since it was a genuine emergency? Can the hospital actually bill me the difference between what my insurer paid and what they charged? And is there any realistic way to get this number down? We're talking about nearly $10,000 on top of what insurance already paid. I'm trying not to panic but this is a lot of money for our family.