This is posted in Illinois so I assume that you are an Illinois resident. Under the circumstances which you describe, it would certainly have been reasonable for you to present yourself to the emergency room of a local hospital. However, it is going to be very difficult for you to prove that you had a medical emergency which prompted you to book a flight to Rochester Minnesota or get in the car and drive for seven hours.
There is no bad faith.
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It sems you may have two available remedies - breach of contract and insurance bad faith. Concerning the contract claim, if it is true that your policy covers the medical treatment, then of course you can sue the insurance company for the policy amount.
However, these cases can be very tricky. The words used in the policy contracts are terms of art - meaning they often have been interpreted by the courts, and thus may not necessary mean what the average person would consider them to mean. For this reason alone, before filing a breach of contract claim, you should at least consult with an attorney who handles insurance claims.
Bad Faith Claims, on the other hand, are typically functions of a given state's statutory scheme. Some of these statutes limit their reach to only certain types of insurance. Given the legal complexity that often arises in statutory interpretation, if you are considering a bad faith claim, you should retain an attorney that concentrates in the area of law.