Is there any way in the Slack API to have the submission button of a dialog open a new dialog or edit the current one?
Not directly. You can use the response_url attached to the dialog submission sent to your server to then post a follow up ephemeral message to the user, containing a button to continue to the next dialog. Once clicked, you'll receive another trigger_id which you can send to dialog.open and start another dialog.
More information on triggers and where to find them can be found here.