I just recovered from a headache after realizing the code given by Google Authenticator on my new phone is different than the one on my old phone. Given how it works I thought they would be the same on both phones. How is it possible they aren't? Dropbox needed the code from my old phone even though I got it to give me the QR code to scan into Google Authenticator on my new phone (I think this is a bug in Droboxes design).
You said, you got a new QR setup code? This invalidates the old one, at least with Google itself.
You should have kept the orginal QR setup code somewhere and used that to set up the Google Authenticator app on the new phone too. Then I would expect the display codes the same.
I have done like this on my various accounts with 2FA, and now have setup Google Authenticator on my 3rd phone without any access problems. I do not use Dropbox specifically however.