Hi, Gaetan,
According to reports on the forum over the last few years, the number of pilgrims on the Primitivo really spikes in August, when most Spaniards take their vacation. When I last walked it, maybe 2011, I even met a couple who had abandoned their Primitivo in August because of crowds and returned the next year in June when I was walking. But I think they may have had the really bad luck of coinciding with a couple of large groups. The offering of albergues is adequate but not huge. That's usually not a problem, but in August you may find a crunch. I'd be surprised if you found a "bed race," though, and Anemone's comments also suggest that shorter stages are a good way to be more certain about a bed. If you get there and find a crunch, you can always call ahead to private albergues (growing number) or pensiones.
IMO, this is one Camino that you should never walk in the early morning dark, because you will miss too much spectacular scenery!
You will love the Primitivo, so please don't be nervous about finding a place to sleep. And let us know how it goes! Buena camino, Laurie