I agree with Opus, only you can deicide it you changed residence, since you plan to move back to NY. That said, I would file as a part year resident of both states.
Taxwise, there's no difference. NY State taxes telecommuters (I'm surprised your employer didn't continue to withhold NYS tax). You pay NYS tax on all your salary. Ohio will give you a credit (either a resident or non resident credit depending on what you call yourself). The Ohio city has already taxed you thru withholding. If you are a resident, you have to file an OH city return. If a non resident, don't file a OH city return, but they keep the withholding.
I'm not fully familiar NYC income tax, but I think you don't owe since March since you didn't physically reside there.
