How to add a signature to an existing record, not a new one?

If you would like to update existing records, you need to use the form formula that allows you to edit existing records.

If you don't want to be using miniExtensions Form as a replacement for the Airtable Form. It's possible to integrate miniExtensions Form with an existing Airtable Form using the redirecting function in Airtable Form to send users to the miniExtensions form URL followed by the record ID parameter.

https://support.airtable.com/docs/applications-of-record-functions#enabling-form-redirects-with-recordid