You don't want to use Relay Domains. You want to use access.db. When you open /etc/mail/access, you will see that localhost and/or the loopback (127.0.0.1) is listed with the word "relay" next to it. You need to add your domain name. For example:
yourdomain.com Relay
If you host other domains on your server, add those as well. Then open /etc/mail/sendmail.mc and add:
FEATURE('relay_local_From')dln
Then create access.db:
makemap hash /etc/mail/access.db < /etc/mail/access
Now recompile sendmail.cf:
m4 /etc/mail/sendmail.mc > /etc/mail/sendmail.cf
These paths may be different on your distro. Don't forget to restart sendmail. When you are done doing this, anybody who has yourdomain.com in the From: part of the address, will be relayed. For this reason, you should look into setting up "pop-before-smtp" to have sendmail use the pop user authentication before is relays mail.