I contacted GoDaddy and they insisted that if I can change the "to" address to an address other than email@example.com and receive those emails with no problem, then everything is working fine on their end. They said maybe that inbox is treating it as spam? I am not receiving any sign of anything getting kicked back.
GoDaddy told me that many people use this relay and they do not have access to the logs. What else can I try? Thank you again, by the way.
Also: I have the email object set to "mail" and character set set to "default". Is that OK?