Hi, Henriette,
My experience in March is that many albergues and other support (cafes, etc) don't open until mid-month, about the 15th (or Semana Santa, whenever that falls). But if you feel OK with longer stages and tending to yourself at mid-day in the food department, it's a lovely time to go--there are fellow pilgrims but no overcrowding.
Of course, the weather can be changeable...but that's spring anywhere. Go prepared with enough warm clothes, rain gear, and a sleeping bag and you'll be fine.
And there threads here about Jan/Feb winter pilgrimages--they definitely require special preparation. A search and/or a PM to
@mspath (an experienced winter walker) will yield plenty of useful advice.