You need to file part-year resident forms in both states. You enter the full amount of the 1099 form income in the federal return. When you do the state returns you will be asked to allocate that income to each state based on where you earned it and more importantly how long you lived in the state.
As a practical matter, you can allocate the income based on how long you lived in each state if it was somewhat evenly distributed during the year.
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