If your temporary move to OH was a condition of your employment or continued employment, you are not a resident of OH and you are still a full time resident of WA. However, you are required to file a non-resident OH return for your earnings in OH, since you physically relocated there for the duration of the work. Your employer should be withholding state taxes for OH while you're there, but they're not required to. So be ready to pay up to the state at tax time if OH state taxes aren't being withheld.

Thank you! The temporary move was not a condition of my employment, rather I was fortunate enough to have the ability to work remotely and stick with my girlfriend while she did her program in OH. This sounds like it falls in line with your comments about "continued employment". As I understand it, my employer is currently not withholding OH state taxes as it was a relatively short term move (~3-4 months). Based on your response, I should plan to file a non-resident OH return and expect to pay state income taxes on wages earned for the duration I was in OH.  

Since your move was not a requirement of your employer, the employer is not required to withhold OH state taxes. But you are still required to pay taxes on monies earned from that WA employer while physically present in OH and earning it in OH. So you'll definitely be filing an Ohio non-resident state tax return.

Keep good records of everything and be able to prove not only what you were paid while in the state, but the dates you arrived in, and left the state.