We just got back from a 3-nighter at Ozette Campground. Not crowded when we arrived on a cloudy Wednesday and backcountry camping was closed, got busier on Friday after they reopened the beaches to overnighters (backcountry permits required!). I was nervous about the 15-site first-come-first-served campground filling up, since there are no other obvious options nearby, but it never quite did. Still, arrive around lunchtime for the best picks.
Moreover, we checked out "Lost Resort" next door, and it seems like a perfectly plausible backup plan, with quite reasonable sites for a private campground.
Didn't see a lot of options for dispersed camping around there, though. In a pinch, you could probably park on Swan Bay Road adjacent to a clear cut. There was one established site there.