My form says I have new submissions, but when I open submissions, there's nothing new there.

There is no new submission besides the last one you see on file, it's most likely a delay, you can mark it as read:

If you haven't setup email notifications, here is how to do it: http://www.jotform.com/help/25-Setting-Up-Email-Notifications, in that way you can tell for sure if a submission is missing in JotForm end. 

Let us know if you need more help, we will be glad to assist you.