If you were a resident of two different states during the tax year (for example, if you moved from one state to another), you'll normally file part-year returns in both states, assuming each state collects income tax and you had income in each state.
TurboTax handles part-year returns, but you need to make sure you've set up your personal information correctly for us to do so. Turbo Tax will ask you about where you lived when you set up your personal information.
Best of luck with your upcoming move.