Your federal refund comes from the IRS treasury department while your state refund will come from your state treasury department. They generally come at separate times.
You can use your State Department of Revenue's website to check status of state tax refund. Processing times vary from state to state and it's not uncommon to receive your federal refund before state; or vice versa.
Accepted in TurboTax just means the government has received your return. This is the status of your return; not your refund. The next step would be for them to process the return and approve your refund. Once they complete that, they update their website with a deposit (or check mailed) date for you.
see below for tracking your state refund:
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