Overseas visitors: California campground availability in May

My family and I are travelling to the US from Europe, and we are thinking of an RV trip in California (or surrounding areas) from the 10th to the 27th of May, as we will be in the area for business. I have realised that my initial fantasy of moving from campground to campground in Yosemite is, unfortunately, just a fantasy, as we'd have to book, and it's likely that everywhere would be booked up already. I also realised that our trip unfortunately coincides with Memorial day and school vacation time, which further complicates things.

Would be possible to still find available private campgrounds or alternatives prior to the Memorial Day weekend (22nd of May)? These wouldn't have to be in typical tourist areas. Would we have to book ahead, or would we be able to go from place to place? What about after the vacation starts?

Thanks