How do I correct state tax filed in Kentucky and ohio when I live in Kentucky but worked in ohio?

KY has a specific form (740 NP-R ) for a reciprocal state refund. Rather than paying TurboTax for KY software, you should try filing that form by hand. It looks pretty  simple.

http://revenue.ky.gov/Forms/2016_42A740-NP-R.pdf

Ohio has a reciprocal agreement, on wages, with all neighboring states. No Ohio state taxes are withheld or due and you do not need to file an OH return. But your home state will tax you on your Ohio income.

If Ohio taxes are mistakenly withheld, you have to file an Ohio return to get a refund. You should ask your employer to stop withholding so you don't have to file an Ohio return every year.

However the reciprocity agreement does not apply to local city taxes. You are not required to file a local city return, but you are also not entitled to a refund of the city withholding (box19 on your W-2), unless there was an error in the amount. If you are subject to a KY county or city tax, you should check to see if the KY city will give you a credit for the OH city tax withheld. KY does allow an itemized deduction (Schedule A) for local tax paid.

It depends.  Chances are you have nothing to correct except with your employer.  Since KY and OH are reciprocal states, you only pay state tax where you live:  KY.  Your original KY return is probably correct.  You might have an amended return you can fill out for Ohio, though, if you are not getting refunded all of your Ohio state taxes (as listed on your W2 box 17.)  You should be getting them all back, because Ohio should not be taxing you.

For your employer, therefore, make sure you fill out Form IT-4NR (link will allow you to obtain it).  This way your taxes will be withheld where they belong:  KY.