State tax filing

Yes.  You can claim a credit in California for taxes paid on the income earned while a resident of California but also taxed in New York.

You can also get a credit in New Jersey for the taxes paid in New York while you were a New Jersey resident.

You should be filing two part-year resident state tax returns for California and New Jersey, and one nonresident New York state tax return.  

In the "My Info" section of TurboTax, confirm that you have entered the correct state information:

1. Click "Edit" to the right of your name
2. Confirm your input under #2 "Tell us the state(s) you lived in"
3. Confirm the rest of the state-specific questions here as well including the "previous state of residence" and "date you became a resident" of the state noted
4. Do the same for your spouse if you are married filing jointly.
5. Go back to the personal information section, and verify the mailing address and "Other state income" section.

Review your input on your W-2 Wages under the "Income & Expenses" section.

Confirm that you have entered the correct state for each W-2 in Boxes 14-15.

This will then allow you to enter your state returns for the nonresident state as well as the part-year resident states.

Prepare your nonresident New York return first.  Pay attention to the information on each screen to ensure the nonresident return is calculated correctly.  After you have entered all of your New York information, you will see a screen like the one below.  Make sure to write down the Allocated NYS tax.  You will need this number to claim a credit for the income earned and taxed in both California and New York.

Next, prepare your part-year California resident return.  You will again follow through the screens to ensure you have allocated only the income earned while you were a resident of California.  You will then see a screen that you can edit to ensure you receive credit for income taxed by New York while you were a California resident.

Finally, prepare your part-year New Jersey resident tax return.  You will be able to get a credit for the taxes paid to New York while you resided in New Jersey.

Essentially, you can get a credit in each of the states you were a resident of if the income is taxed twice.  This will avoid any double taxation.  The input will depend upon the state reflected on your W-2, Box 15 for state source wages.