To send luggage ahead to Santiago, your best option may be Casa Ivar:
Early April will coincide with Holy Week (5/Apr - 12/Apr 2020). That may put pressure on accommodations in places since there may be a surge in pilgrim numbers. There will also be more tourists in major cities for the Holy Week celebrations.
Are you starting in Seville? You might need to book accommodation well in advance for Seville or Caceres during Holy Week, because there are no dedicated pilgrim hostels and there will be a flood of tourists for Holy Week. (The hostels in Seville and Caceres are backpacker / youth hostels). In Merida and Zamora there are albergues exclusively for pilgrims.
On public holidays like Good Friday and regional / local fiestas, you may also find that the stores are closed. If in doubt, ask locals what to expect.
Climate-wise, I think it's one of the best times to walk - the landscapes are green and you won't face the searing heat of the summer. But of course, the weather is unpredictable. In 2019 there was a week of heavy rain in the south of Spain - an exceptional weather event. Even if you don't expect that, you should be prepared.