Chapter 59.18 RCW (Residential landlord-tenant act) does not require pro-ration of rent except in specific situations. A tenant simply wanting to move generally is not one of those specific situation.
A written agreement often has a provision that all the terms of the agreement, except the rent period and amount, would remain the same when the rental changes to month-to-month. You should review your lease to see what you agreed to do.
There is no law that requires a tenant to physically occupy a rental to the end of the month. The RLTA may require the landlord to do the move-out inspection. Thus, "showing up for inspection and accepting keys on the move-out day" does not necessarily mean that your landlord did anything wrong.
You should review your specific lease and facts with your attorney to find out your legal options.