How much does it cost to amend my state returns after its been accepted?

If you filed with TurboTax, no, generally it doesn't cost any more to file an amended return.

• However, if the amended change requires an upgrade to a higher version of TurboTax, you'll be charged. The reason for this is, had you initially filed the form or change, you would have needed to upgrade and paid for the form or change at that time.
• For example: you filed using the TurboTax Online Deluxe version but now you have investment income on Form 1099-B. To report investment income, you would need to upgrade to TurboTax Online Premier.

Wait for the IRS to finalize your return before sending in an amended return. 

• See How to amend (change or correct) a return you already filed?

Generally, Form 1040-X,  Amended U.S. Individual Income Tax Return, must be filed within three years of the date the original return was filed, or within two years of the date the tax was paid, whichever is later