Just adding my two cents --
My 18-year-old daughter and I just returned from walking the Frances from Sarria to Finisterre. While the Camino was quite crowded from Sarria to Santiago, and we sometimes felt a bit like we had joined a foot-race with more veteran, much stronger and at times rather competitive fellow-pilgrims, we never found ourselves without an albergue bed at night.
Taking our cues from the vets, we rose early (by 5:30/6), generally put in 5 miles before stopping for breakfast, and generally arrived at our destination by 12:30 or 1:00 when the albergue opened.
We mainly stayed at public albergues, but we did notice quite a number of private albergues and small hotels available as well, especially through Santiago.
I know the numbers are expected to increase in July and August, and that may worry potential pilgrims. Hearing reports of crowding on the Frances, Caity and I almost decided to walk the reportedly beautiful and much less-traveled Ingles instead. But finally we decided to stick with the original plan/dream, be open to whatever came our way, and trust that "The Camino Will Provide". We are so glad that we did -- because among those Frances crowds we met amazing people, shared wonderful conversations and meals and experiences with folks of all ages, and really soaked up the joy and camaraderie of the Camino.
Good luck -- and Buen Camino!
Cindy & Caity