Can I transfer submissions from one form to the next?

Can I copy over the responses from the earlier import, which did include the responses to the new form (see form that shows OLD).

There is no built in feature to transfer submissions from one form to the next. However, we do have a import tool that you can use to import submissions from your previous form into your new one. 

Please reference the below link for the Import App that you can try:
http://apps.jotform.com/app/import_data 

First thing you would need to do is export the submissions made to the previous form (http://www.jotform.com/help/44-How-to-Export-Form-Data-to-Excel) and then use the App provided to import the submissions to the new form. Do let us know if our assistance is need further.

Regards