After some more testing and research it doesn't look like the hosts file is relevant to this problem. It is a sendmail issue. I've found information indicating I need to create a "generics table" that will handle the reverse mapping of the users. I first modified my sendmail.mc file by adding the following lines:
FEATURE(`genericstable', `dbm /etc/mail/genericstable')dnl
I then created a file /etc/mail/genericstable and added the reverse of what's in the virtusertable:
makemap hash /etc/mail/genericstable < /etc/mail/genericstable
I created a /etc/mail/generics-domains file:
I added domains to my local-host-names file:
I then created the sendmail.cf file:
m4 /etc/mail/sendmail.mc > /etc/sendmail.cf
and restarted sendmail.
No luck so far. I'll keep at it, but if anyone can see something I'm obviously doing wrong or missing from my description, please let me know!