Hi worried -- If you have proof that you have correctly informed your local council about the change in circumstances and they have failed to act resulting in these back payments accruing, this would be an "official error" caused by their failure to act accordingly. In this situation they are supposed to waive the charges, as the error has been caused solely by their failure to take the appropriate action.
If you are still liable at the end of it all, then you can make an arrangement to repay the arrears at a weekly amount you can afford without causing you hardship, they have to give you that chance.
As for payments at your new address, either ring, write, or call in asking for a DD payment to be set up or sent to you at the new address from the date you moved in.