So, interesting solution to the problem.
The relay was bad. However, it wasn't completely none functional. So, likely over the last few days the relay has been sending variable power instead of consistent power.
So, while the bulb itself worked, the installation of the new relay provided too much consistent power to a weakened bulb that was already fraying, on the element.
The problem likely would have been fixed with the relay replacement, however, there were two different bulbs installed. One was a Sylvan, the other was some generic no name bulb. That bulb is what what blew out with the new relay and was apparently frayed the most due to the originally faulty relay.
I was told that this problem is also often associated with a burn out bulb at the license plate. If your vehicle has a plate bulb, a burnout of that bulb can mess with your headlamps.