If your total gross taxable income is really only $12,000, you would not owe income tax because it is less than the standard deduction (unless you are married filing separately under some circumstances.). But as noted, if you are a US citizen, you are required to report your world-wide income and pay US tax, no matter where you are living. Whether you also owe foreign income tax on the same money depends on the laws where you move. Also, as a US person, you must declare ownership or control of foreign bank accounts on a separate web site from your tax return.
Regarding NYS, the question is whether or not you are a resident of NY. If you are not a resident, you don't owe state taxes and don't file a state tax return. The rules for residency are discussed here.
https://www.tax.ny.gov/pit/file/pit_definitions.htm#domicile
Basically, if you maintain a domicile in NY, you might be considered a resident. For example, you keep a small apartment, you visit a few times a year, you maintain your doctor, dentist, and other significant business and social relationships in NYS. Even if you abandon your domicile in New York, you will need to file a part-year resident return for the year you move out of state, that reports and pays tax on income earned during the portion of the year that you were a state resident.
