Yes, you can delete your State return and start over in TurboTax Deluxe and up.  Simply click Upgrade on the left menu to see products & pricing.  Here are some other options:

1. You can Clear & Start Over in TurboTax Online as long as you haven't paid or registered.  
2. If you have paid or registered and it won't let you, you can navigate back to the beginning and simply edit any information you would like to change.  
3. You can Create a New Account by using the same e-mail address, just under a new Username.