Can I amend my State returns now even though it was just accepted earlier? Do I have to wait for 2 weeks like the Federal return?

Amend will not be available in TurboTax until late in March.  When you amend both the state and federal returns will be amended.  If you only need to file the amended state return you can chose to do that after you amend.

You generally must file an amended return within three years of the date you filed the original return or within two years after the date you paid the tax, whichever is later.