You can advise the employer, but since in California you can be fired for no reason (unless you have a written contract or union membership that provides otherwise) it probably won't make a difference.
Unless you are in a unionized environment, correcting the employer's records won't help you very much -- your employer can fire you for any reason, at any time, or for no reason at all. When and if your signature becomes an issue in a contested legal matter, you can protest the authenticity of the signature.
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