This website is in Spanish, but you can easily see how many "albergues" (i.e. pilgrim hostels) there are per stage from Sarria onwards (just use the arrow at the top saying "siguente" to go to the next stage). If you click on the name of the albergue and look for "disponibilidad", you'll see the months it's open. "Todo el ano" means open all year - I've randomly clicked on a few, and it seems a good number are open all year.
Just follow the arrows into town, and you'll be led straight past the albergues. If the municipal one is full, the hospitaleros will direct you to the next one along the way.
I just looked at the stats from last year, and in the month of March 6,400 people finished the Frances, with 4,500 of those starting in Sarria. That's around 200 people per stage, per day. So don't worry, you'll have plenty of company!