Manchester State Park (SP) in Port Orchard looks like the closest SP campground. They have water and electric, but no sewer. However, there is a dump station near the entrance; so you could just dump your tanks once or twice a week, a minor inconvenience. Perhaps your brother can drive through and check it out for you.
July is in the peak summer season, so if you decide to stay here, you should make reservations in advance (they take reservations nine months ahead) to make sure you have a spot. Again, your brother could let you know which sites are the best. However, please note that Washington SPs limit stays to 10-days, maximum, in the peak season.
We stayed at Dash Point SP (northeast of Tacoma) a month or two ago, and if Manchester SP is anything like Dash Point, you'll really like it there. Also, send me a PM if you'd like the itinerary for our 78-day roadtrip up the PCH/Hwy-1 from San Francisco to Seattle, and back. There's lots to see and do on the Oregon and Washington coasts, and you can't find a more different landscape from Phoenix. Compared to our four inches of annual rainfall in Arizona, on the western side of Olympic National Park in the Hoh (River) Rain Forest, they get 14 to 17 FEET of rain per year; and it doesn't look anything like Flagstaff!
Note: You may want to save one night for the Washington Land Yacht Harbor (Airstream) RV Park near Olympia. If you've never stayed at an Airstream Park, it's an interesting experience; beautiful grounds and dog park, within a few blocks of a Costco and WalMart where you can refuel and restock for the trip home. It's about seven miles east of the junction of the 101 and I-5, and about a mile south of the freeway on surface streets.
Manchester State Park | Washington State Parks and Recreation Commission
Camping Rates | Washington State Parks and Recreation Commission
Port Orchard (Washington ) - Campground Reviews - RV Park Reviews