Can I file my state tax return now and wait to file my federal until later?

A state tax return cannot be e-filed unless the federal has been filed and accepted by the IRS.   So, no, you cannot e-file the state first.   

No, you must E-file your Federal Retune prior to E-filing your State.

• The TurboTax state program is designed to transfer information from your federal return. 
• You can however, print and file the state separately.