Part-year state returns are complicated. One of my partners said he told clients they could move only on January 1.
The general rule is that earned income is taxed only in the state where it was earned. However, the a state may take into account your overall income, including income sourced in another state, in determining the tax rates that apply to the income being taxed by that state. NY will not tax your earned income from before the move, but that income may cause you to pay more tax on the retirement income you receive after the move.