I filed my federal tax using H&R Block, but not my state because they tried to charge me. Is there a way to just file my state tax using Turbotax?

No.  When you use TurboTax the information flows from the federal return to the state return.  You cannot use TurboTax to prepare a "stand alone" state return.

You could try going to the state's tax web site to get the forms and prepare your state return by hand.