We walked last year, starting from Pamplona in early May. We were in a massive wave that we couldn’t escape for weeks. And we walked slowly, so any wave should have passed us by, but it didn’t. The problem persisted for most of our Camino.
We didn’t care where we stayed, private albergues or not, private rooms or not, just as long as they took reservations because we were sending a bag. We often stayed at off-stage places, but it didn’t matter. Sometimes I called as many as ten places in various towns before we could book two beds. I remember several times calling every single place listed in Gronze without any luck. We eventually learned that we were okay as long as we were booking 4-5 nights out. That limited our flexibility which we didn’t like, but walking in May was lovely. And the crowds didn’t bother us apart from having trouble finding beds.
We did meet a few other pilgrims who got frustrated and quit because beds were so tight, or who took a bus ahead many miles to try to escape the wave we were in.
One tip that I would share is not to depend on booking.com for what is available. There are often beds or rooms that are not listed on booking.com if you call the albergue or pension directly. Speaking enough Spanish to make a booking on the phone will help. Also, other people who are walking alone and just show up at large public albergues that do not take reservations might be able to find beds. But that won’t work as well if you are a couple and not at all if you are shipping a bag.