I would recommend the Queets, but I have to warn you that it might be too low in August. Unfortuately, this is true of all the Olympic rivers, and most rivers of the northwest: August/September is the dry season. But sometimes the rain comes early, so it’s not out of the question.
If it is runnable, the Queets is a phenomenal river for packrafting. You have to ford (or paddle across) the river to access the trail that takes you about 15 miles upstream, where it disappears into the rainforest. The ford means there’s far fewer hikers than you find on the Hoh, so it’s more of a wilderness experience. Most of the year it’s too high to ford, so when I’ve run it in the winter/spring I’ve had the entire river/rainforest to myself.
The Hoh is a magnificent river, but it tends to have more dangerous timber hazards than the Queets (I’m speaking of the National Park section of the Hoh, where it’s paralleled by the trail, which is what you’d want for packrafting. Downstream from the NP it’s not so dangerous, but you’d be walking along the road or arranging a shuttle, and you wouldn’t be in the pristine ancient forest). Like the Queets, it tends to be too low in August.
If the rivers are all too low, one option might be to hike along the coast and use your packraft to explore the mouths of creeks at high tide. The outlets of those coastal streams offer some truly enchanted forested gorges to paddle into for a bit. But to be honest, in August (unless it’s raining a lot) I’d be more inclined to leave the packraft behind and head for the magnificent high country: Skyline Divide, Bailey Traverse, etc. Then come back in winter or spring, and paddle the rivers when they’re running nicely, and you have the place to yourself, and the forests are dripping with moss and gloomy grandeur.