Income is taxed first by where it is earned. Where you live may have an obligation, in your case, both cities of residence. You will need to file with the correct wages and tax to get a refund from Fairfield. Without a corrected w2 form, you may end up having to show proof of your move at some point. I remember Fairfield was pretty fierce years ago when I lived there. It may be worth talking to them in person. You can file online.
File Fairfield at Income Tax Division | Fairfield, OH.
You will need to paper file with Franklin. See Welcome to Franklin, OH Again, file the correct income and tax. They will be thrilled to get more than shown on your w2. You may want to make a copy and take it with you to Fairfield or to be ready when they start asking questions.
Good luck!
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