2 Factor Authentication screwed up my iCloud login...

I recently enabled 2 factor authentication on my iPhone. However, I was not able to verify my Mac. I logged out of my iCloud so I could log back in again, thinking this may have been the issue. I also changed my password, just in case. When I go to the System Preferences menu to login to my iCloud, it asks for my Apple ID and password, which I enter. From what I understand, after I hit the "Next" button, it's supposed to take me to a screen where I can enter my verification code that I receive on my phone. However, no such field pops up. Instead it loops backwards, and asks me for my password again as if I had never entered it to begin with. I don't get any error messages, or incorrect password messages, or device disabled messages. I have been able to successfully login to iCloud.com, so I'm pretty sure it has to be something with the Mac. Does anyone have any thoughts on this?