Doh! It turns out this was not a Drupal issue, but a Centos issue with SELinux. I was able to check SELinx settings by running the following commands

> getsebool httpd_can_sendmail

which returned

> httpd_can_sendmail --> off

and

> getsebool httpd_can_network_connect

returned

> httpd_can_network_connect --> off

I enabled them by running this
> setsebool -P httpd_can_sendmail 1
>
> setsebool -P httpd_can_network_connect 1

And the Drupal SMTP module worked beautifully after that. Hopefully this helps someone else as well.