Get your taxes done using TurboTax

In your case, you need to file a nonresident return for CA to recover the withholdings. On the nonresident return, declare earnings of $0 so you can get a full refund for the State income taxes your employer withheld. You should also tell your payroll department so they fix this in the future while you are still living away from CA.

1. So yes, you can get your state income tax withholdings back. As long as you enter the state income tax withheld from your W-2 into TurboTax, you will get credit for paying that and it will flow to the CA state tax return. (in your TurboTax program, click on the left panel "Wages and Income", and edit the W-2 you entered to review your entered state withholdings)
2. You can't claim SDI refund on your tax return. If your employer doesn't refund you, you have to file a claim for a refund with CA Franchise Tax Board. Go here for instructions: Help with refunds 
3. For years 2018-2025, there is no longer an income deduction for moving expense reimbursements, this may be the IRS law the moving assistant company is referring to. 
4. If you have to pay back the lump sum for moving expenses, there is no way to get a refund for the tax that you paid.

If you make sure your residency is entered correctly, TurboTax will guide you to be sure you get your CA state income tax refunded. 

This is how you should answer the residency questions for CA.  This will ensure you file a non-resident CA Tax Return with $0 CA income. When you get to the part to prepare your CA State Tax Return, when you get to this section below: