When you login to webmail, regardless of if you're logging into cPanel first and then clicking on the webmail link, or going to example.com/webmail in your web-browser they should be attempting to connect to the same webmail service.
Have you possibly recently updated your domain name's DNS settings? This could possibly be causing your browser to try to send your mail credentials to the incorrect server.
Unfortunately without knowing your account information there isn't much we'd be able to do from our end for you. You might want to submit a ticket with your domain name you're having issues with, as well as any email accounts or webmail clients that are giving you issues logging in with.
If you feel comfortable posting that information publicly here, we'd also be glad to take a look into it for you.
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