IRA Contribution with Federal extension but not state extension

What state? Every state program is different.

Since you are using the CD/Download TurboTax software, here is something you could do that will work for any state, but it has some disadvantages, which I have listed below.

File your federal and state tax returns from different TurboTax files. Save a copy of your tax return with a different file name. In the copy, remove the IRA contribution and complete the state return. File the federal return from the version that includes the IRA contribution, and file the state return from the version that does not include it. But you probably will not be able to e-file the state.

Disadvantages of this method:

• Somewhat complicated
• Possibly confusing
• Will probably require filing your state tax return by mail
• May cause confusion next year, because no matter which 2022 file you transfer information from when you start your 2023 tax return, either the federal return or the state return will not be what you actually filed.

26 USC 219(f)(3) states that such a contribution is deemed to have made on the last day of the preceding taxable year, so seems proper that the deduction is included on your state tax return.  

some states do not give a deduction for IRA contributions.

It has nothing to do with having requested an extension of time.