Lake Pleasant Park is just north of Seattle, and likely you best bet. There are two parks in Issaquah but you are almost ON the freeway.(I-90) There is also an older, small KOA in Kent, near a Boeing facility. Those are your choices in or near Seattle. Definitely not very RV friendly.
Seattle is wonderful to visit but not so much in your MH. All built one hills and zero parking downtown to stop and sightsee. There is a ferry from Bremerton that you can walk on that takes you to downtown Seattle at the waterfront. Most things to see are uphill from there.
See Mt St Helens, Olympic National Park and Mt Rainier National Park when in Washington if you can!