There are two kinds of digital certificates (IDs): those that are issued by a reputable Certificate Authority (they provide complete PKI functionality) and self-signed certificates (IDs). I have a hunch that you are using the latter. You can create a new one in Acrobat. You do not need to delete the old one.
The simplest way is to start signing process and when you get the "Sign" dialog click on the triangle next to "Sign As" field and select "New ID". It will guide you through the process of creating a new digital ID. You can also get to this dialog from the "Security Settings" dialog in the preferences. How to get there depends on which Acrobat version you're using. In Acrobat XI it is Edit->Preferences->Signatures->Identites$Trusted Certificates->More. Then select "Digital IDs" and click "Add ID"