Can I file separately if I lived and worked in CA 6 months and moved to NY 6months and worked 5 months here? Is it better to file a certain way in this situation

For federal income taxes, you are going to file once, with both W-2's. For each respective state, you will file on the money you earned while employed in that state, so you will file your state taxes in both states.

More specifically file your federal return using your current address (ex. Ny resident). File a nys tax return as a resident and a ca as non resident

Actually, you should file your California and NY returns as part-year residents.

Further, provide a copy of the other state return when you mail in each return; that way, they know how the total income has been allocated. It answers questions before they are asked and resolves problems before they even begin