My controller is now stuck in this Sign Up or Sign In mode.

Neither operation works because the controller is not connected to Wi-Fi, and there is no option to do so.

How do I get out of this?

Hi, we're sorry for the inconvenience that it caused. Have you tried to restart your remote controller? Have you tried to swipe down from the top of your screen to see if that would provide you an option for wi-fi? You may also send us a screenshot showing the problem for further assistance. Thank you for your support.