How do I know if my state taxes have been filed? It is not showing a return date and is only asking me to “continue” with them. My federal is already submitted.

No, if your state status is "Continue" you didn't complete and submit your state return yet. 

Log in to your account, and select Continue under State taxes. Follow the on-screen instructions. Keep going until you submit the state return in the File section. 

If I follow the prompts, it takes me back to the homepage. What do I do?

If I follow the prompts, it takes me back to the homepage. What do I do?

Try closing TurboTax, clear your Cache and Cookies, and reopen TurboTax.  Make sure you have ran all updates.