I wouldn't be concerned. I walked the second half of the Norte in August last year just staying in municipal/parochial albergues until Arzua, but it should be a bit quieter when you go. I can think of several albergues that were officially full by about 4 or 5pm in the afternoon (and Miraz much earlier). But nearly all of them had strategies for coping with this - places like Soto de Luña opened up a whole hall the same size as the dormitory and provided gym mats. Aviles was full by about 5pm but the hospitalero was giving people details of other places to stay in town and that was quite typical. I can think of a few places that were unstaffed where people arriving late just bedded down in the hallways, but I think that's more of a crazy August thing when, with the hordes and the heat, the rule book seems to go out the window...
The three main pinch points were Cadavedo with 12 beds, but people walked on a few km to a camping chalet place I think. Ribadeo only has 12 beds which is most disproportionate to the size of the town, but we walked on the 7km to Vilela which had about 30 or so beds and was fine and gives you a head's start into the hills. Miraz has much larger places on either side of it at Baamonde and Sobrado, but they sorted something out for us...
Ach, you'll be fine