Tantlu, considering how the tourist industry has suffered, you might consider "approaching" some folks who ordinarily rent out apartments through AirBnb/VRBO. Private duenos might consider a better deal than they advertise for several reasons: 1) Said beat up tourist business 2) Your long-term stay 3) Being the lone renter might mean less wear and tear on their property. We have rented the same apartment 3 times but, sadly, it is no longer available. Being close to a metro station is worthwhile as is having a washing machine, which would be a huge money-saver for you.
Oh, and by the way, renting a room in a home might weary you after a while. It's a crap shoot as to how comfortable you will really be sharing a place with others. We did this for short stays of 2-3 days in Granada and in Valencia on previous trips to Spain and determined that apartments were the way to go for us, a retired couple. While "immersion" is a great goal, remember that you need to be comfortable and fully rested so you can concentrate on your studies.
Good luck,
Elizabeth