Hi Lost_but_happy,
"This" domain is the one you received the error message. As I said I have checked the DNS records for this domain, you can change your account from @xxx.onmicrosoft.com to @thedomain.com, and then use this account to access Lync Online, or send/receive emails via @thedomain.com in Outlook to see if it works well.
Thanks,
Bourne Zhang