It's poor form to reply to your own post, but I fixed this and thought I'd report back to let folks know what I did.
I'm using the RA virtual appliance, so I enabled Webmin and took a look at what needed to be updated on the server. I ran the updates that were available, fixed the subsequent MySQL connectivity issues that resulted from the updates, and tried again. Lo and behold, everything worked. With a few changes...
Password - re-entered the same password as above (copy & paste)
Port - 465
That's it. I thought it might be the Linux Firewall but no. All outbound was allowed by default. So there must have been an update that was needed or when I thought I checked port 465 yesterday, I just needed to re-enter the password. I'm sorry I can't pinpoint the exact thing that fixed this, but the reality is that there shouldn't have been an issue to begin with.