,do your charitable donations on your federal income tax automatically transfer to your state?

You don't need to re-enter your charitable donations for your state return and if you are able to use the deduction to your benefit then TurboTax will automatically apply it.

But, each state calculates tax liability differently from the Federal and it's possible it may not help you for a variety of reasons.  Some states have higher standard deductions, which means you might not have enough deductions to change your tax liability.  Some states have more limited itemized deductions and may not allow charitable donations as a deduction.  And, some states don't allow any itemized deductions at all.  It's also possible you may just not have had any more tax liability to offset with the deduction.