We thought we renewed our plan, but we are out of jotforms. Help!

Yes, you have currently a bronze plan that allows to have up to 25 active forms. Your account was created under such limit was implemented and your existing forms were grandfathered. 

You will need to disable/archive/delete some of your old forms in order to have more form for new forms, disabling/archiving forms will not delete the submissions data. 

You may also want to have a backup of your data and these guides will help you doing so: 

https://www.jotform.com/help/73-How-to-Download-Form-Submissions-as-Excel-CSV-PDF

https://www.jotform.com/help/356-How-to-Download-the-Submission-Files

https://www.jotform.com/help/374-How-to-Export-All-of-Your-Data-at-Once

You may also check your account usage on this page: https://www.jotform.com/myaccount/usage 

I hope this helps.