Generally, you CAN pre-book PRIVATE albergues, as well as hostals, hotels, etc. Many (but not all) can be found on
www.booking.com. As suggested above DO use other sources, like
www.Gronze.com to locate other properties not on booking.com.
From Madrid to Segovia, you should be okay finding pre-bookable lodgings. From Segovia up to Sahagun, there is just not a lot of infrastructure. You need to be more flexible. This is NOT one of the more popular Middle Age pilgrimage routes, like the
Camino Frances or Camino Portuguese.
However, once you reach Sahagun, you are joining the very heavily used
Camino Frances. Especially at this time of year (Easter marks the beginning of the peak season) you need (IMHO) to pre-book whenever possible, and DO SHOW UP!
When on Camino, especially along the Frances, if I must e-mail ahead to request reservations, I compose a note in English on my iPhone, then translate it to Portuguese or Spanish using the Microsoft Translator (the smart phone version of Bing) or Google Translate, both on my iPhone. I always provide the Spanish (or Portuguese) and then the English original in my e-mail so they can figure out what I meant to say if the translator returns less than spot on results. Lately, I have been finding that the Microsoft Translator is returning "better" translations. Caveat Emptor...
The Camino Madrid was established more or less "recently." The first third or so, follows the old cattle road the ranchers used to move their cattle herds from low to higher pasture as seasons changed. After that, it resembles the same sort of rolling, up and down hills you would find in Rioja and eastern Castilla y Leon. In many places as you walk along farm roads, you parallel the AVE train route into Leon.
The Camino Madrid is a worthy route, and it is beautiful once you are out of the valley where Madrid is located. Once you reach Segovia, the scenery is markedly better, IMHO.
I started out from Madrid in April 2016. Ironically, on my fourth Camino, I had to stop at Sarria (so close yet so far) due to arthritis-induced, bilateral, hip, knee AND ankle pain. This was made worse by rain for 10 straight days, snow at Cruz de Ferro, and generally miserable walking conditions for someone with arthritis. By the time I humped it into Sarria, the walk was just not fun anymore. So, having three Compostelas already, I (wisely I think) pulled the plug and rode a bus to Santiago. There I was pressed into service as a volunteer at the Pilgrim Office for five days.
I hope this helps. I wish you a healthy and wonderful Camino Madrid.