I figured out the problem. While I did re-enter the password in the RetroArch app, I also had to add the password manually to each RetroArch .cfg file I have. Like RetroArch.cfg, NES.cfg, SNES.cfg, ect. I also had to remove the chevos_tolken from each of the .cfg files and then all games launched from Launchbox logged into Retro Achievements.
Will I have to do this everytime I update either Launchbox or RetroArch? This is going in an arcade cabinet, so that might get pretty annoying.