I walked the Norte, leaving Irun in mid April. Its true that the Norte is rainier on average than the Frances, but its also warmer, being at a lower elevation. The majority of days start and finish at sea level, and the weather was quite agreeable. We dodged our share of rain drops on a few days of course, but that year was much drier overall than my first outing on the Frances, where it rained for several weeks. Of course, the second time I walked the Frances (all mid April departures, mind you) was sunshine and flowers the whole way.
I would not be worried about the Norte being excessively rainy. Every year is different, and you get what you get. Worry about the relentless up and down for the first ten days, I found it to be a substantial physical challenge!
As for the Primitivo, I cannot comment, although I would be surprised to find anything other than remnants of snow at elevation by the time you get there. Cruz de Ferro is higher, in the neighborhood, and snow free at that time of year.
Buen Camino