Thank you for your question. As a test I created a new email address in your cPanel, and was able to send a test message from webmail successfully, without being logged out.
Be sure when you log into your webmail at your domain: example.com/webmail, that you are using the full email address as the username:
I would need more information to replicate the problem, such as the browser type and version you are using. What URL you are accessing your webmail from.
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