I've seen exit row availability (and more seating availability in general) on Saver fares on flights departing <24 hours before departure, and I've actually made one of those last minute bookings over July 4th and successfully kept my seat assignment. Personally, I'd usually avoid picking up Saver fares because there are basically no guarantees other than the fact that they fly you from point A to B and you get free bags checked if you're elite - no changes or cancels unless AS makes a significant change to the schedule on your ticket on their end.