I found out the issue. Alienware updated their AWCC on 30 May, and mine was updated yesterday. I fully uninstalled AWCC and was able to get in fine. I reinstalled AWCC, and the same problem occurred. I have removed AWCC for now, and am able to fully access the game again.
Whenever I try to launch the game, the game launcher just states, "Connecting to Account Server." The GameClient.exe just sits there as a black screen. Any ideas, or anyone else having this issue?