Olympic National Park

On your way to or from Mt Rainier you may want to spend a couple days around Port Townsend.
Either at Fort Worden State Park, my first choice or if they are full, Fort Flagler State Park. You can take a whale watching tour from Port Townsend up into the San Juan islands for Killer Whale sightings.

We stayed at the Port Angels KOA, when we visited Olympic NP. It appears that they have 22 full hookup sites.

You'll have a much easier time if you can prepare yourself to avoid hookups. My favorite place is South Beach, about 3 mi south of Kalaloch.

I like the country over by Lake Quinalt and Forks. Lower elevations are not quite as wet. The Hoh Rainforest is very interesting for people to see.

IMO you’ll need to accept some dry camping to really experience ONP. Proper equipment makes that very comfortable.

Olympic NP is huge and is surrounded by US-101 on its W, N, & E sides. On the east side we have stayed at Lake Cushman with hookups. It is run by one of the Tribes. On the north anywhere in the Sequim, Port Angeles, Lake Crescent area is good so you can explore that area of the NP. Lots to do and see...very scenic. We like Salt Creek Campground (Clallam Co. Parks) at Joyce just west of Port Angeles. Hookups with a sewer dump station on the way out. On the west side google the Forks/LaPush for Campgrounds. Kalaloch campground is right on the coast, run by NP, and very popular.

I don't think there is a 'best' side because you cannot drive thru the park to the other side. You'll need to camp at sites on both sides since you have to drive all the way around the place. We stayed on the east on the hood canal at a very nice place. And on the west at Forks (I never heard of the place but my wife knew all about it.) which was very soggy and rainy. You'll want to see Hurricane Ridge, Hoh and the east side. Not as much in the south so far as I know.