With your limited time I would go with the North side of the Olympic NP.
Take the Edmonds to Kingston ferry North of Seattle so you don't have to take the long drive around.
You can stay at a lot of different places, but
Accommodations at Lake Crescent Lodge | Olympic National Park & Forest WA has a lot of the things you were hoping to see/do.
Plenty of hikes and vistas to behold. Enjoy!
p.s.--you picked a great month to come. August is even better.