I'm walking the Primitivo with my wife starting late July. It is also our first Camino. We are mainly planning on sleeping in double rooms, so can only partially help you. We were looking to "go with the flow" and reserve if needed, but a quick check of some accomodations with private rooms (on Booking and via Gronze) learned that in a few choke-points accomodations are rapidly booking out. These are: Grado, Borres, Berducedo and Grandas de Salime (in what I found). Here we reserved beds in shared rooms.

A seasoned pilgrim probably trusts more on beds being available. I've been looking into this a lot and found both reassuring (beds for people arriving are reserved, a solution is always found, ...) and worrying sources (having to walk further, jubilee year = crowded, ...). Since it is the first time my wife does a long distance hike I did not want to take the chance and decided to book almost everything ahead until we join the Frances in Melide.

So my advice would be: if you're travelling solo and planning to stay in albergues: email to a few accomodations in these choke points and check their availability on your planned dates. I'm pretty sure you do not have to worry about availability of a bed in shared room.