How to troubleshoot nonexistent domain name when using our domain name
You get error messages that the domain name is nonexistent when using Method 2 of your domain name, where you have created a unique 3 node domain name like lists.xyz.com.
Email error message looks like this

Step One
Verify you have set up the DNS correctly at a third party site like MX Toolbox or DNS Stuff.

Step Two
Send an email to your List from a third party site, for example gmail.com or yahoo.com.
If you can send email to your 3 node domain name from off site, then the problem is internal.
Check that your outgoing mail server is pointing to valid DNS servers, to look up the domain name of outgoing email.
Optional
You may need to add the new domain name's MX Records (lists.xyz.com) to your local DNS server. This is so your outgoing mail server does not get confused with the entry for xyz.com
This is especially true for Microsoft Exchange servers.