Book an apartment or home
Most rentals are cheaper than hotels. You either go out for dinner, or cook your own meals. Both options are no problem at all in Saas-Fee.
The landlord or rental company will provide a free SaastalCard upon arrival. The pass allows free local traveling by bus/cable cars, and discounts on several activities. If you own a rail pass, you get free bus and train traveling in a much wider area, and discounts on mountain transport.
Interhome offers thousands of holiday homes and apartments across Switzerland. They offer discounts on both early and last minute bookings. That makes the Interhome website a good place to start your search:
https://www.interhome.com/switzerland/: all holiday homes
https://www.interhome.com/switzerland/early-booking-discount/: find the discounts for early bookings
https://www.interhome.com/switzerland/last-minute/: find the last minute offers