Will it be very busy at this time?
The busiest months on the Francés - if you look at the statistics - are normally May, June & September. Before I left may 22nd this year, I was also a bit unsecure, whether I would always find a bunk each night, because I was going to walk in May and June this year.
I never had a problem finding an albergue. I usually arrived between 2 and 4 p.m. I seldomly stayed in any major stop of the Camino which
Brierly or other pilgrim guides prefer. In small pueblos I never had a problem finding a bed.
If you want to walk late into the afternoon, after 4 p.m., I'd think about booking in advance. Otherwise I don't think you'll have problems finding a place to stay, if you're flexible and your budget includes a stay in a pension, hostel or hotel once in a while.
If you actually don't find a place to stay, there'll always be the possiblity to take a taxi to the next place which will cost about 1 Euro a kilometer. Since the stops on the Francés between pueblos are usually not more than 10 to 12 kilometers apart, this will probably won't bust your budget, if that happens once in a while.