No. Don't file your state tax return until you have the ITIN. States don't create or approve ITINs, so they won't know what to do with your return or documentation.
The IRS will forward your federal tax return for processing after the W7 is approved and the ITIN is issued. Your federal return is being mailed, and you won't efile the federal return.
When you get the ITIN in 6-10 weeks, go back into Turbo Tax, update the SSN field to include the ITIN, and then transmit the state return by itself.