Cannot log in with Apple ID after performing password reset

To anyone still needing help, this is what was done to fix this issue.

If you reset your iCloud email and password online that does not update your icloud account on your computer with the new email address.

You need to login to your iCloud account online and set your primary email back to the original one (your computer is still using this email) reset the password if you would like.
Now restart your computer and you should be able to log back in.
Once you are logged in, go back to iCloud website and change back to your new email.
Before you log off your computer go to Setting>iCloud and remove the old iCloud account and add the new iCloud account.

That should be all you need to do.
Now your computer is using the new email and you should be able to login using your new password.

After talking with Apple support for a couple hours, booting into safe-mode, and re-installing the OS (at their behest) we ended up just going into recovery and using the "resetpassword" command from the terminal.