You will need to check with the Washington State Department of Licensing (DOL) to find out if you can apply and obtain a WA driver's license while your previous state of Michigan has your license suspended there.

Likely WA will require you to get your MI driver's license reinstated before you can acquire a Washington State license.

Washington laws so state you have 30 days to get your Washington driver license once you establish residency in Washington State however the DOL states to do so you need to turn in a valid out of state license, which your suspended Michigan license would not be.

Also keep in mind that your driving record from Michigan will be brought over to WA when you obtain a Washington state license.

Revised Code of Washington (RCW) section 46.020.091 subsection 3 states that if a person previously licensed in another jurisdiction applies for a Washington driver's license, the department shall request a copy of the applicant's driver's record from the other jurisdiction. The driving record from the other jurisdiction becomes a part of the driver's record in this state. So it would appear that the reason that your MI license was suspended will follow you to WA.

Again, to find out for certain if you can apply and obtain a WA driver's license at this time you should check with the DOL. If you must reinstate your MI license before getting a Washington license then you will need to contact the MI SOS to find out how this can be done so that you can again be licensed to drive.