Drat, the CNAME trick won't work because my Domain won't respond to foobar.mydomain.com. Apparently, that's a limitation of Apache's virtual hosts.
In other words, if your browser shows foobar.mydomain.com, and it's not a frame that's hiding the real site, then foobar.mydomain.com must be a domain that's been created in one way or another at DreamHost.
 Continuing this thought, I believe that it is possible to configure httpd.conf's Virtual Host for your specific domain to respond to www.yourdomain.com, yourdomain.com, and *.yourdomain.com, but I don't know if the panel can implement such a change. I know it can be done manually, though.