OH has no provision for allowing W-2 employees to deduct job ("business") expenses. As others have said, the feds no longer allow job expense deduction, starting in 2018. If you live in an OH city, with a city income tax, a job expense deduction may be allowed on the city (but not School District) return. You'll have to check with your city to see if they allow "form 2106" (job expenses) deduction.
WA does not have an income tax, so the question of paying tax to WA is moot. But, even if the company you worked for was located in a state with a tax, you would not owe tax to that state, if you never physically worked there*. Pennsylvania, Nebraska, Delaware, New York and New Jersey are exceptions to that rule. see: http://www.journalofaccountancy.com/issues/2009/jun/20091371.html
*Would you owe tax to the employer state, if you occasionally made trips there? Yes, but "nobody" bothers with it. See http://money.cnn.com/2013/06/28/pf/taxes/business-traveler-tax-threat/ If you live in a state without an income tax (e.g. FL or TX), it’s more likely you should file in the work states. You can't use the "it all comes out even" rationale for not filing.
