No, you can leave the state field blank in TurboTax if it is blank on your forms. All income from W2s that is taxable by your resident state will transfer over automatically as part of your federal adjusted gross income.
As for the prize winninngs, generally yes, you would answer "Yes" to receiving money from another state in the "My Info" section. Depending on the state there may not be a tax return filing requirement (income too low for example). Each state has it's own filing requirement for non-residents.
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