How can I retrieve archived forms/submissions?

Also, we have been notified that we have surpassed the free limit of forms.  How might we delete some of the forms (5) and still keep the submissions?

There is no way to delete a form and keep the submission data active in JotForm.  You can export your data to a format such as Excel:

https://www.jotform.com/help/44-How-to-Export-Form-Data-to-Excel

This will give you a copy of your form data to keep for your records. 

Submissions for archived or disabled forms can be viewed the same as for active forms:

https://www.jotform.com/help/269-How-to-View-Form-Submissions