My federal return was accepted on the 22nd and I have already received it. My state return was accepted on the 27th but it has not yet been deposited. Is that normal?

Federal and state tax refunds are issued at different times from each tax authority, IRS and a State.

If accepted by the state use this TurboTax support FAQ to check the state tax refund status - https://ttlc.intuit.com/questions/1899433-how-do-i-track-my-state-refund