1. If you received the lump sum while you were a legal resident of CA, it is CA income. You were a CA resident until you established your domicile (your fixed, permanent home) in WA. If you received it after you became a legal WA resident, it is WA income.
2. Expenses related to a move OUT OF California are not a deductible expense on your California return. WA does not have a personal income tax.