Hey Adam,
Sorry to hear about these problems. I’d first like to point out that on your Downloads page here on this site, you can manually download the 3.11 update and install it if that ever becomes urgently needed on your site.
In the meantime, however, I know this sounds like simple thing to try, but does anything improve if you delete the license key in your site settings for now, then “Disconnect” that site’s URL here on your account License Keys page? Once that’s disconnected, then head back to your site settings and try adding the license key again.
See if that makes any difference – if not, then unfortunately I’d recommend deleting the license key again and “Disconnecting” that site URL again, then manually upgrading to 3.11 and trying to enter the license key on your site again.
Thank you!
George