I've walked the Primitivo in August a few times--so I don't think you'd be mad at all! It's a beautiful time for walking, and I met many wonderful folks to walk with--and a great sense of community. A few of the public albergues have overflow space (which may just be a gym floor), like the one in Cadavo Baleira (I slept in the gym while staying there in 2011), and during the busy summer months the albergue in Grandas de Salime also has overflow space.
A few new albergues have opened or expanded in the last few years, (Grandas de Salime, Padròn, and La Mesa have added beds or mattresses (though La Mesa is small and still a place that fills easily) , and there are new or fairly new albergues in La Espina (private), Berducedo (private), Castroverde (Municipal), and As Seixas (Municipal and Private).
When I walked the route last year in August, the places I remember filling up were Bodenaya (with the next albergue being only 1km further in La Espina), Borres (surprisingly, since I don't believe that the albergue in Campiello 3km before was full. The two extra people chose to sleep outside), and La Mesa (although Berducedo, 5km before was also not full). In every case that I recall, the pilgrims without beds were able to find a sleeping situation that worked for them--sometimes with help from the hospitalero.
We were also told by the hospitalero in Bodenaya, that just two days before us the albergue had been only half full. So, the business can vary by day as well. If you find yourself in a particularly busy pack of walkers, you can often slow or speed your itinerary a bit to get away from the peak of pilgrims.
I wouldn't be overly concerned, unless you have specific issues or are particularly wary of being caught without a bed. If that is the case, I'd wait to see how busy it seems when you arrive, and then in any of the places you are concerned about, you could look into reserving a bed ahead of time. For instance, if you're concerned about the mountains, you could reserve a bed at the private albergue in Berducedo--or just plan to stay in whichever albergue has space there (the municipal or the private), rather than walking on to La Mesa.
Hope this helps!