If I file a Certificate of Amendment to change my corporation's name, do all my employees have to sign new employment contracts?

No, this is just a contract modification that you can give notice of. Your corporation under its new name is going to assume all the obligations of the contract that the previously name corporation undertook, and the employees are going to continue working under the existing contract, so nothing is changing except the name.

You should send written notice to the employees (using the notice provision that's in their contracts) giving them formal notice of the new name and the effective date of the new name.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.

No, a name change does not affect the contract, as the entity continues with the new name. You may want to inform each of the employees under contract that this has occurred but it is not necessary to sign any new contracts.