A Microsoft digital storytelling app for creating and sharing interactive reports and presentations.
I don't know if this helps you or not - if I sit and continually hit refresh (slowly so I can see whether it responds), my sway will eventually connect - both signed in and not.
This suggests to me that some servers in your farm are indeed responding so it feels like something has or has not been propagated across the server farm.
I know that certificates have been mentioned and one user mentioned an expiry message on 28th June - the day the issue started. It might be worth checking that new certificates are correctly installed on all servers and are correctly related to the certificate on the SSO servers.