If I am filing with TurboTax, can I send my federal taxes on earlier date and state taxes a couple of weeks later?

[Kat]

You can file your federal and state returns separately.  However, if you're e-filing your returns, you won't be allowed to e-file your state return(s) until your federal return has been "Accepted" by the IRS.

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[KurtL1]

You can elect not to file the state return when you file the Federal return. You would then log back in and file the state return when you are ready,

On March 20th the IRS extended the filing deadline to July 15th. If you need to wait you can file between now and July 15th.

Edited 03/30/20