Walking the Primitivo in Sept.

I don´t know the albergue occupation in September, but extrapoling my experience in June, you won't have any accommodation problems. Unless, in Grado, Salas, Tineo, Pola, Berducedo,Grandas, Fonsagrada, Cádavo, Lugo and San Román there are also cheap hostales (20 -25 euros, in some of them breakfast included). After San Román you will enter the Camino Francés in Melide and I don´t dare to extrapolate.
In September you will have very few muddy places, but as a disadvantage you'll find some fields and meadows not green but yellowish.