You are correct that you will file a resident tax return for the state of Arizona and non-resident tax return for Illinois, but you will not be taxed on the same income in both states.
It is important that you complete your tax returns in the correct order to ensure that the state taxes paid credit will be calculated correctly.
1) Prepare your federal tax return and have it pass an error check
2) Prepare your nonresident state (Illinois) next
3) Lastly, prepare your resident state (Arizona).
For additional information, please review the TurboTax article Multiple States—Figuring What's Owed When You Live and Work in More Than One State.
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