In TurboTax there are allocation worksheets for the states that as you go through the screens you will see when you set up your information that you were a part-year resident of two states. You tell the program what income is taxable for each state.
No, both states are not going to tax the income.
Generally the deferred income is taxable to the state where you lived when you worked to earn the income.
Here is an article with more information Strategies for Managing Your Tax Bill on Deferred Compensation
You may also find this article helpful Multiple States-Where To File