I haven't heard of Ohio (or any other state) exempting travel nurses from state taxes. Try googling it (I found nothing).
You will have to file an OH non resident return.
Ohio does a convoluted tax calculation for non-residents/part year residents. It calculates tax on total income, then it calculates a non resident/part year resident credit, which it subtracts from the tax it calculated on the total income. The credit is calculated as your non-Ohio income divided by Total adjusted Income multiplied by the total tax. TurboTax (TT) does this by allocating your income as either Ohio or non-Ohio. W-2 income will be allocated by the state name abbreviation shown in box 15 of your W-2. TT will ask you, item by item, in the state section, how much of your other income is Ohio or non-Ohio income. Make sure that your non-Ohio wages show TX (Other state postal abbreviation) in box 15 of your W-2 screen, with the TX amount in box 16.
Since TX does not have an income tax, boxes 15-17 may be blank on your actual W-2. At the W-2 screen, in TT, enter TX in box 15. Put the box 1 amount in box 16 and 0 in box 17.
