If we moved out of our house and told the utility company to turn off the electric but it failed to and kept billing us, are we responsble for the bill?

You should not be responsible for the bill if you properly closed your account. The most important thing to do is to 1) check all your bills, statements, agreements, etc. with or from the utility--do they specify how to close an account and, if so, did you provide the proper notice or documentation? and 2) check to see what evidence you have of having closed the account properly (e.g. copies of a letter; proof of delivery; etc.).

Legally, as stated, if you closed the account, from and after that point, you should not be reponsible for any bills. If there was a specific method or process for closing the account, though, you would have had to have done it in accordance with that process; if you failed to follow the right procedure, the utility would likely be entitled to keep the electricity on and in your name until you do.

Practically, even if you acted correctly, if you can't prove that, you would have a hard time prevailing.

Assuming you did properly close your account and feel you can prove that, first provide copies of the documentation to the utility, to see if you can't get them to resolve the matter. If they will not, you then have two choices: either to simply not pay, wait to see if they take legal action, then defend yourself on the basis of the proof that you closed the account; or to affirmatively bring a legal action of your own seeking a "declaratory jugment" (court determination) that you do not owe the  money.