For income tax purposes, it doesn't matter which state you are a resident of, since neither Alaska nor Texas has any state income tax. It does seem that you are a resident of Alaska, and I assume you use your Alaska home address on your federal tax return. When TurboTax asks for your state of residence, you should select Alaska.
I can think of two situations where it might make a difference in your federal tax return if you were a resident of Texas rather than Alaska. But since you "work and live in Alaska" you should still select Alaska as your state of residence.
(1) If you have health insurance from the Marketplace ("Obamacare") and you qualify for the Premium Tax Credit, the amount of the credit might be different depending on which state you are a resident of.
(2) Texas is a community property state, and Alaska has optional community property. If you are married, the differences in the community property laws might make some difference in your federal tax return, especially if you do not file jointly.