I’m having exactly the same problem as this on TSO Host, who claim that all the ports are unblocked. I’ve also checked with Office 365 that their IP hasn’t been blacklisted.
One things I’ve done is modified the plugin to add “gethostbyname” function calls around the entered value, which then means the IP address is translated and I can see it in the logs, this has been useful in getting me slightly different messaging, but still nothing that works.
In your case, GoDaddy block SMTP pors (25 and 587) and you have to send email out through their relay…
https://stackoverflow.com/questions/21841834/phpmailer-godaddy-server-smtp-connection-refused