Legal Question in Employment Law in New York

No, they are not required to pay you for the additional two weeks. Providing 'two weeks notice' is a courtesy that employees provide to employers and which employers rarely provide to employees. New York is an at-will employment state and your employer can dismiss you at any time for any non-discriminatory reason, including providing two weeks notice of your intent to leave.

I advise clients to forgo a two-week notice unless they have been with a company for at least five years for precisely the reasons you have cited. Any less than that and the courtesy is unwarranted and unnecessary, in my opinion.