If memory serves, you need to set up masquerading. I believe the "DM" directive in your sendmail.cf file, containing your domain name. Be aware, though, that if you're shoveling out a large amount of email, you may be flagged as a spammer by other mail servers, and they'll blacklist you.
Google can point you to some things, as well as some threads on here, too:
Of course, you have to know what to look for, before you can search for it.
Sendmail and/or postfix have their own particular kind of black magic involved with them, but for good reason. They WORK, and work well. It's worth investing in a book or two on the subject.