If spouses live and work in separate states, you have to check each state for the filing requirements. For both New York and Georgia, you have the option to file a state joint or separate return.
Because your husband is military, there are special rules. The most important one that may apply is that active duty military do not change state residency automatically when they are stationed in another state. See What is my military state of residence? - TurboTax Support
If your husband only had active duty pay, you would only have to file a New York Resident return, unless he abandoned his New York state domicile and established in Georgia. You can file New York state only for 2024, and his military pay is exempt from state taxes. Note: if he had other employment in Georgia, he would have a nonresident filing requirement.
Your husband will have to contact DFAS to have the Georgia state withholding corrected. If he did not intend to change residency, you can file an amended return for Georgia 2023 to indicate he was a nonresident and you will be able to get the taxes paid refunded. He will have to file a 2024 nonresident return as well to get the withholding refunded.
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