If you acknowledge it, then you are bound by 12 months as the contract stated, even if they do not have a copy of it. The reason why they are asking you to "resign" is because they do not have a copy, and, an employee handbook non-compete mention may not be binding upon itself if it is not specifically signed.
Now, even if the non-compete is signed, then it may or may not be binding depending on how reasonable it is. See HERE. There are two ways to find out if it is reasonable:
1) Simply engage in the work. The risk is that the old company will sue for the breach under the non-compete, in which case one may argue that the non-compete was not reasonable and let the court decide; or
2) File for a declaratory judgment prior to engaging in this work . This is a type of case where one asks the Court to clarify the rights/duties of parties under the law for a contract. The positive is that this has one be the Plaintiff, so there is less risk of being seen liable as no work has yet been performed. The con is that if one loses and the court upholds the non-compete, one may be liable for the old company's legal fees.
You do not have to sign the document. However, they also do not have to pay you the severance pay. But if you already acknowledge that you signed the original non-compete, then this may not matter if the non-compete they are putting in front of you mirrors that of the original.
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