My advice is to not postpone your trip because of Easter. Rather, embrace it! Don't avoid Easter; plan for it.
Yes there will be accommodation issues - mainly in the larger cities and neighbouring areas during Holy Week (the week leading up to Easter Sunday), especially the nights of Thursday April 9, Friday April 10 and Saturday April 11. So figure out where you will be on those nights. If it is Seville, Guillena, Zafra or Merida, book something immediately on booking.com. Maybe plan an extra day or half day to watch the activities.
I'd expect "everything" in Seville to be booked, at a high price, well in advance, but Easter in Seville is supposed to be fabulous.
Even in small villages, you can often see Easter ceremonies that add to your experience. If you are walking anytime from April 5 to 12, keep an eye on the accommodation situation a day or two ahead.