I don't think you'll have too much trouble finding lodging. Just be flexible - if the first place you try is full, go to the next one, and maybe walk to the next town if you are striking out - often the people running the alberques will call ahead for you to see of another place has space if you choose to walk on. Also, in addition to alberques, there are casa rurales and hotels available - more money than a bed in an alberque, but if you have the budget, they are a good option.
A lot of people follow the stages in guidebooks - Bierley is a popular one - so lodging in the towns at the end of each published stage tend to fill up. One method to avoid that is just to stop in other towns other than the ones at the ends of those stages.
Also, if you are finding places are filling up, then perhaps stop a little earlier in the day - walk until 2 instead of 3 or 4, for example.