I will be walking for 10 days this May (starting a bit earlier in the month). It is her first Camino. We are starting from Villafranca del Bierzo, for what its worth. I wouldn't worry too much about it being too busy. The busy season for the stretch from Sarria to Santiago is July and August. It shouldn't be too crowded when you walk it. If you are really concerned, you can book your private rooms in towns or villages "between stages" (that is, between the towns that the popular guide books choose as stage end points). I did so this past July, the busiest season, and had plenty of time when I was walking in solitude with no other pilgrims in sight and very very few times when there was anything approaching a "crowd". But before going that route, I would humbly suggest that you not dismiss the opportunities to meet and interact with your fellow pilgrims so quickly. Many find the interactions with their fellow pilgrims to be a highlight of their Caminos.
I can't speak to an organizing company. I've always booked my rooms, when doing so, myself. Apps like
Wise Pilgrim and Buen Camino or sites like Gronze.com (in Spanish, but easily translated by your web browser if you don't want to practice your Spanish) have excellent listings of accommodations options, including private rooms, and contact information for them. The
Wise Pilgrim app will even compose a WhatsApp message in Spanish asking for a reservation on your behalf.