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For income tax filing it is best to always use the name that is currently on record with the Social Security Administration (regardless of what the W-2 or other document may have)
You may file in Georgia as a part year resident and either married filing joint or married filing separately. From page 9 of the Form IT-511 Individual Income Tax Instruction Booklet: A married part-year resident or nonresident with income earned in Georgia whose spouse is a nonresident with no Georgia source income may file either a separate return claiming himself/herself only, or a joint return claiming total allowable deductions. If one spouse is a resident and one is a part-year resident or nonresident, enter 3 in the residency status box and complete Form 500, Schedule 3 to calculate Georgia taxable income.
See Form 500 Individual Income Tax Return (Fill in on-line, print and mail)
I hope this helps.
Best wishes and congratulations on your marriage!