I did all my Primitivo with sunny days. There was mud in some places, but easy to pass except for the stretch before Tineo (500 mts?) (by the way don't try to avoid it going through the surrounding meadows, you could have problems to return to the Camino !).
If the rain is going to reduce the beauty of this Camino. Who knows, sometimes, the mist associated with rain increases the beauty of forests and mountains.
Let me give you a piece of advice: In case of rain, fog or strong winds do not do the Hospitales variant, even if your family Camino is going to do it. The Pola de Allande variant is also very nice, but more protected by the surrounding mountains. In case of rain, after passing Puerto del Palo you better walk the road till Montefurado.
Sorry, I don't know anything about albergue conditions because. I always was in hostales (20-25 euros with breakfast included, San Román 17, Berducedo 35).
Buen Camino.
Bo Camiño