When I sign a form can the signature stay for the next form?

I'm not quite sure if I understood your concern correctly. Could you please provide us more information on how your process flow works? Would you like the response or submission be editable so that the signature would stay? Something like a multi-step form? You can pass along as the edit link of the submission for that. Here's a guide about it: https://www.jotform.com/help/40-How-to-Let-Users-Update-Their-Form-Submissions-at-a-Later-Date 

Example:

1. Form has 3 signature widgets,

2. Applicant first fills out the form and signs the 1st signature widget.

3. The edit link for that submission is then sent to the HR, the HR reviews the submission then signs the 2nd signature widget.

4. The edit link for that submission is then sent again but this time to the Department Manager, he/she reviews the submission and signs the 3rd signature widget. 

To send specific email notifications with the edit link on them, you can use conditional logic, here's a guide about it: http://www.jotform.com/help/167-How-to-send-to-a-specific-email-address-based-on-a-selection 

Please do let us know if I misunderstood your concern.