Mystery solved. Though a bit embarrassing, perhaps this will help someone else out in the future:
As mentioned, we have two Exchange servers (version 2013). They're in a database availability group (DAG) for redundancy and performance. HAProxy on a Linux box acts as a load balancer for the DAG.
Before the load balancer was in place, we would switch the NAT rules on our firewall from one server to the other in order to test each server, essentially simulating what the load balancer would be doing once in place. When the load balance was put in place, everything still worked well, except one thing: our anonymous SMTP relay.
Some of our other servers use SMTP to send messages and notifications. One or two of them only allow a simple server address and port when pointing to an SMTP server, and do not allow for more settings like user/pass, which could be used to setup connection to an authenticated relay. So, we had setup anonymous receive connectors (not an "open relay") which were scoped to only allow the servers that needed access based on their IP addresses.
Well, when the load balancer got put in place, the SMTP relay wasn't relaying traffic for our other servers that were trying to connect to it any more. This was because the load balance changed the source address of SMTP request that pass through it to it's own IP. Well, I quickly and foolishly added the load balancer's IP to the anonymous relay scope, and WALA!...our servers could use the relay again......AND SO COULD THE REST OF THE INTERNET!
So, contrary to what I had be concered about in the original post, no one had our credentials, nor had anything malicious been installed on our servers, I had simply accidentally made our anonymous relay and open relay, all because I allowed the IP of the load balancer to be part of the relay scope.
My issue now is finding out a way for my other servers to send email using a relay that is based of off IP address and not authentication (the way I had it originally), since their software does not allow the latter. Of course I could just point them to one of the Exchange servers directly, bypassing the load balancer, but that would defeat the purpose of load balancing between our two servers. Any ideas on that? Can the load balancer somehow be more transparent and not change the source address of IP packets that pass through it?