You have to file a New York non-resident tax return to get a refund of taxes paid there in error.
You may not be doing anything wrong. Since you didn't have taxes withheld for Georgia for one month, it is likely you will owe taxes to Georgia for that month. Since you are getting the taxes back from New York for the month they should have been applied to Georgia, that explains why you owe Georgia taxes, as the $287 refund from New York represents the bulk of the taxes you owe to Georgia.
[Edited 3/3/25 at 10:02 AM PST]
@cpmcilwain
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