According to the Washington State Department of Licensing, you are legally obligated to title and register your car in the state and get a state driver's license within 30 days of establishing "residency." Their definition of that includes the following:
-Arrive in Washington and
begin establishing a home in the state.
-Register to vote.
-Receive state benefits.
-Apply for any state license.
-Seek in-state tuition fees.
WA State Licensing: Moving? Get WA license
So I think that if you DO end up buying a home in Washington, you should quickly re-title and register your vehicle(s) and get a WA license. So long as you're not receiving state benefits or registered to vote in WA then I think you should be fine, because you are not technically "establishing a home" in WA yet, especially if you're not going to be employed in WA while living there. But if you are going to employed when you are living there then that might be a question to call or e-mail the WA DOL about. I know in California, you have to register/get a license, etc. 30 days from gaining employment in the state, and here in Oregon if you live here at least 6 months of the year and/or pay resident income taxes then you could be considered a resident and must go through the whole license/registration process. It really varies.
You might also want to ask in the Washington state subforum.