First of all you should get a signed copy of the one month notice you already gave. Otherwise you can't later prove that you did give the notice.

If you already have it, don't worry.

But did you read the other articles in the contract too? I have seen a signed contract saying that termination notice is 3 months but you still have to pay at least 3 months or all remaining months whichever is higher. I of course didn't check the law if such clauses are possible, but you should fully understand what you have signed.

If the contract really says 1 month notice and nothing more, than get your notice paper signed and don't worry. Even if they cash your cheques, you can sue them and get everything back.