Having started twice on May 13th i experienced a good variety. Yes, mornings can be busy. Yes, ther for sure will be talk about full albergues and whatnot. I opted for unconventional stages, always taking alternative routes if one presented itself, sometimes walked long into the afternoon. I had my fair share of "crowds" but also a surprisingly high amount of solitude. Once had to walk into the next village for a bed, but otherwise no issues finding a place to sleep.
Theres a popular recommendation to book SJPDP till Pamplona, since most people walk pretty much the same stages, and i think that can be a good idea. Once in Pamplona, there are more options and you have a better feeling how busy it really is.