You are correct that your military income should not be included in your Non-Resident New York return, only the income earned and taxes paid there.
If you are below the New York filing requirements, you are also correct that you would not need to file a return, unless you paid tax there that you want refunded.
If you do file a Non-Resident New York return, prepare this return first, before your Resident New Hampshire return.
Here's more info:
https://www.tax.ny.gov/pit/file/nonresidents.htm
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