You return does automatically save while you are working on it. 

You may not see the My Account drop down menu if your return is not open, in which case you can just exit out. If your return is open you should be able to find My Account on the top left (see attached for additional help).

I cannot sign out either.  This is so frustrating.  What is going on???

Please try clearing your cookies in your browser and then log back into your TurboTax software.  Many times, this will help clear the issue. 

Instructions to clear cookies to restart TurboTax 

@joblo