If you're using shared iPads, you will get to a point where no more students can sign in, because there is a maximum amount of students who can be signed in to Seesaw on one iPad using their Google account.
The issue is actually a Google issue, not a Seesaw issue, so to solve it you need to do the following:
Go to https://accounts.google.com/ in the browser.
And/or, you can also click Sign Out, and you should see an options to remove all signed in accounts: