No can probably cannot get your California state tax withholding refunded from you employer. They usually are not able to do that. But speak with them just in case.
You should file a DE 4 with your employer to stop California withholding until you move to the Golden State in November and a G-4 to start GA income tax withholding.
A California nonresident will only pay state income tax on income earned while present in CA. See "Residency and income sourcing" in COVID-19 frequently asked questions for tax relief and assistance.
Georgia
Not withholding Georgia tax is not illegal since you did not file the required paperwork with your employer to have tax withheld. As a GA resident, you are liable for GA tax on all your income, regardless of where it was earned until you move to CA. The longer you go without GA income tax being withheld, the more you will owe next year. You can make an estimated tax payment if you wish to make up for the missing withholding.
