Tacoma Power's Mossyrock park
This campsite we went to in the fall was beautiful and not over crowded. We happened to get one of those glamorous PNW dry sunny weekends when we camped here and it was so nice to sit around the campfire.
Very family friendly and pet friendly, not a lot of privacy at the campsites but enough space to have your own set ups. I would recommend choosing a site on the river side because the views were incredible and I think those have more privacy than the interior sites.
Price is very average and the park is clean and well maintained. They even have cute painted mushroom tree stumps throughs the trail around the park.