How do I save changes after editing a submission?

I see the option to edit a submitted form...i.e. if you want to change a persons address or other info- but there is no save button? 

There is no save button. You will have to re-submit the form to save the changes. When you re-submit, you'll get an email with a prefix EDIT in the subject line. The autoresponder email, if present and if the Send on Edit is enabled in its settings, is also sent to the submitter. 

Screenshot of the Send on Edit option:

Does that also mean that a new payment will go through? My form involves a one time or auto renewal option but they are through PayPal and required.
Sent from my iPhone
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No, editing submission will not process any payment. The payment field is disabled on the edit submission link. 

If you need to change their subscription, you need to ask them to re-submit the form. You can cancel/refund their first payment.