So I have actually spent 2+ weeks on a November tour of the Olympic Peninsula and we loved it, way less crowded and the weather was fine. But we did get lucky with the weather, that storm that rolled through was pretty epic (friends who live in Bellingham are still underwater). There are some park alerts:
If you do get out there, we really liked Kalaloch, I bet those campgrounds are never empty but they weren't too bad, if you stay for a few days you can slide from one site to a better one at changeover time, some of them have spots you can walk to and sit on the bluff:
(and yes, we drove from campsite to campsite with our top up, hah, I'm sure we're not the only ones). IRRC they might have shut the water off for the year (or maybe I'm misremebering) but come with full water. The beach at Kalaloch is HUGE, so you can definitely get away from the other people (and most of the folks in the campground never seemed to get down onto the beach). With no one very close to you be sure to follow that leash law obsessively
We stayed at the Hoh Rain Forest campground at the end of the Hoh Road and it was very cool. There's a small loop hike to do (no dogs this time) that was short enough didn't feel bad about leaving the dog in the van, got to see some serious moss action! Also saw Roosevelt elk. We were the only vehicle in the campground.
I will second SportyGma on the Mora Campground at La Push... very cool spot, the beach is rockier and not as big as Kalaloch but we got to hang out and watch people surf, and the campground itself is under big canopy feels pretty protected even when it's blowing and raining.
If the weather totally shuts you out from that side of the park you can head for the rain shadow, on our lap we also really enjoyed our time up by the Elwah Dam site, Port Townsend (you can camp at the marina on the point) and Port Angeles / Sequim / Dungeness Spit (great hike out the spit, no dogs though). Looks like some snow up on Hurricane Ridge but looks like the road's open.