I want to help, but you didn't specify Office 365 or webmail. However, I will go with Office 365, currently our most popular email solution. It's unfortunate you are unable to contact us, have you tried all the options at support.godaddy.com ?
In any case, if you are using Office 365 email, Microsoft offers this good troubleshooting index with some test tools that may be of use.
Also, the fact that you can communicate within your own email suggests maybe a settings issue where "the world" cannot see or communicate with you. Again unless you check it with our online help articles, this is something only support could identify.
I hope that helps,
Thomas D. - GoDaddy | Community Moderator 24/7 support available at x.co/247support