After getting in contact with EA Support through a web chat, they helped me find a solution to both EA App not launching and EA games not launching through Steam, changing my DNS to Google Public DNS fixed both issues and I now have access to all of my EA games. I'm not sure why this worked, since I had no problems in the past with the previous DNS settings I had, but I'm just glad it's working now.
****EDIT: While the App launches, it crashes after a few minutes, and then will not launch again untill I restart my PC, then another crash after a few minutes. Any EA Games that require access to the EA App to work (most of them, inlcuding ones on Steam) will also crash after the EA App crashes. Due to this, I still do not have access to my game on EA App AND Steam. Please fix your stuff EA.