We stayed in private rooms and found booking a day in advance was plenty of time. On a couple occasions our first choice wasn't available, but there was no trouble getting an alternative. We did not stay in any albergues, but some of them do have private rooms with private bath, so that's another possibility to look into.
If you enjoy researching, though (and we do), no reason not to start looking through some guides and online sources now to get an idea of what's out there -- everything from 5-star international luxury to cozy inns run by a local family. Gronze.com is great for online listings of all accomodations. And Goggle maps was very helpful in seeing exactly where in town, or how far off the beaten path, places are.
Larger inns and hotels may have websites or have reviews on TripAdvisor. Some of the smaller places may have nothing but a phone number and, occasionally, a Yelp review. "Pot luck" is all part of the fun.
There were a few places we did book before leaving home, but only because we wanted to stay in specific hotels. As it turned out, we could have walked into any of them on arrival and gotten a room. But that was in October. A different time might have a different outcome, so if you have your heart set on a place, go for it.
One really fun surprise of all of this is that we started running into a handful of the same fellow pilgrims night after night. No pre-planning, hadn't seen them since breakfast that morning, but lo and behold.... !! Lot of laughs and camaraderie -- it's all part of the Camino.
Enjoy your planning, enjoy your walking, and then enjoy your memories for a lifetime.