You can allocate your investment income based on where you lived when you received it. When you go through the part-year resident return (link has instructions), TurboTax will ask you how much of the total investment income belongs to Georgia. Enter the amounts you received from Jan - May, while you were a resident of GA.
Allocating unearned income is pretty straightforward: just allocate it to the state you were a resident of when you received it. There are some examples in this TurboTax FAQ.
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