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.
