When you prepare your New York part-year resident tax return you will be asked how much of the interest and dividends is New York income. Any interest and dividends that you received after you moved to New York, or that was credited to your account after you moved, has to be allocated to New York. So your New York income will include part of the income shown on the 1099-INT and 1099-DIV forms. There is nothing to file for Texas because Texas does not have personal income tax.
It doesn't matter where your bank and brokerage account are, or what address is on them. What matters is where you lived when the income was paid to you or credited to your account.
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