You will have no problems finding private rooms along that section. What would be helpful for you is a good route map detailing the towns and distances between them. I used the Michelin No. 160
Camino de Santiago. It is tiny (6.5 €) but loaded with detailed information, i.e Albergues and services in every single hamlet, town, and village along the Camino. I talked to many people who were using Booking.com to arrange for the next day lodging. Many, many private albergues on Booking.com. The service is free because it is the owner of the establishment who pays Booking to get their place advertised at their site. Another source of lodging info along the Camino is the brochure from Jacotrans which lists dozens of private lodgings from where they pick-up/drop-off. Even if you don't use their services, the info still relevant. A great source are local Offiuces of Tourism; they know of the private rooms at "Mama & Papa" which enhances the experience tenfold, if you ask me... Much like you, private bathroom was not a must for me. Sadly, often pilgrims leaving at dawn will take all the toilet paper with them and any supplies provided by the house. Some were the same ones who looked down on those staying at the private rooms-- obviously, because their behavior was so exemplary...
I did not use a guided service, always found a private room. To be honest, the Camino has got to be the best marked, well documented long distance trail in the world to do on your own.