I did not change my address with my employer because I work from home & didn't want my employer to try to pay me less based on my new location of GA.
You are a part year resident of both states. You will need to file a Georgia Income Tax return and pay the taxes for your wages earned during the time you lived in GA. You may also have to pay a penalty due to not paying your state taxes on time through withholdings.
It is a good idea to change your address with your employer so they can withhold taxes for the state.
In the Personal Info section of TurboTax, enter GA as your State of Residence, and indicate that you lived in another state - FL - during the tax year. This will prompt the program to produce the correct tax forms.
Legally you became a resident of GA for tax purposes on the date you began living there with the intent of making it your new home. It doesn't matter if you don't yet have a GA drivers license, voter registration, etc.
Whether you changed your ID or not, if you lived and worked in GA in 2023 you will have to come to grips with the fact that you will owe GA tax for the part of 2023 during which you lived and worked in GA. Your employer may or may not be willing/able to withhold GA tax from your paychecks, but now that you live in GA, you are subject to GA state tax. If the employer cannot withhold GA tax for you for 2024 you should start making estimated quarterly payments to GA yourself to avoid a big tax bill at tax time next year.
https://ttlc.intuit.com/questions/1901227-how-do-i-file-a-part-year-state-return
And...you will want to explore the laws in GA for how long you have to obtain a GA ID or driver's license, etc.
https://dds.georgia.gov/georgia-licenseid/new-licenseid/how-do-i-class-c-license