Generally, you pay taxes where you work and also where you live.
Your description suggests that you will file part-year resident returns for both New York and California (you were living in New York State when you were laid off, right?).
Both your regular wages and your severance are New York source income, so add them when entering the New York interview.
California will likely give you a credit for New York taxes paid. This appears on line 43, Form 540 (but you are filing form 540NR as a part year resident, so look for a line that is reporting tax paid to other state.
On what form was the severance reported? A W-2? Then just enter it like the other W-2.
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