The ambulance company did not file my insurance, even though I gave them the information and they agreed to file it.

You may be able to prove that you don't owe the money, on the basis that you relied, to your detriment, on the promise that the ambulance company would seek recovery only from your insurer. Arguably, you should not pay the claim and if you are sued, you should assert "waiver" or "estoppel" as an affirmative defense.