I just changed my Apple ID password, and then had the same problem as above. I kept trying to sign in, then got as far as entering the verification code, and it would just revert back to the original sign in screen. I did a hybrid of the suggestions above and it worked. Ie, on my TV screen I entered Apple ID. Then entered password.
However, the hybrid step I used was to go into my new MacBook (2017 version), click on the little apple symbol, get to account then security, and found the screen that showed that I have two step verification. Then I clicked on a button that said to send me a verification code for a new device. The new code popped up on my MacBook. Instead of using the verification code that was sent to my iPhone (as I was doing for the last 30 mins) I entered that new code that I requested via my MacBook. It actually worked.
Forgive me if I sound very low-tech. That's because I am (embarrassingly) low-tech. But maybe this will help someone who's having the same issues. Cheers.