If you are at a full campground, chances are very high that someone in the campground will be willing to share with you.
So ride past the camp-full sign with a smile and wave at the ranger, and look for someone who isn't using much of the site and is hanging around outside. Ask if you can split the site and the cost with them.
I usually do NOT tell a sad story of how screwed I'll be if I can't find a spot / they don't share, because you want to leave them a polite way to refuse your request. There's a fine art to asking for help on bike tour, and it's really easy to manipulate people to get what you want, but it doesn't really feel good and it's not really fair to the benefactor/vicitm.