Is there a way to delete the files uploaded with forms without deleting the form itself?

You can delete your submissions for gaining upload space again. You could integrate your forms with Dropbox or Google Drive, at that time, you will be able to save the uploaded files even if the submission is deleted.

Moreover, there is an application called Auto-Delete Submissions. This application deletes new form submissions instantly as soon as they are received.

If you have any question or issues, please do not hesitate contacting us. We will be glad to assist you.

Thank you for your reply Beril. 

However, we have it integrated with a google spreadsheet as well. Will deleting the submissions also delete entries within the spreadsheet when it is synced?

No, the submitted data will be deleted from the JotForm database only, the data will be sent to your spreadsheet normally, also the uploaded files will be sent to your integration with either Dropbox or Google Drive. 

So you may use the auto-delete submissions app and you will receive your data in your integrations, if this is not the case, do let us know and we will be glad to help you.