I'd add Pontevedra, A Coruna and Betanzos as possibilities in Galicia. The smaller towns of Cambados and Combarra are worth a quick look, and Tui just north of the Spanish border.
Farther east there's Leon (though it's far enough inland I guess it could be pretty hot). Oviedo is a rather quiet city with several very old churches. Then there's the beautiful Picos de Europa area. Burgos is another good stop, but it's also an inland destination, so likely not to be cool as the coastal regions. You might even make it as far as the Basque Country since your car will be an efficient way to move around.
Pick up a book or two that covers the are comprehensively to get an idea of all the options.
I didn't have massive problems finding lodging in that part of Spain during my 2016 trip even though I didn't book rooms very far ahead. San Sebastian is very popular and could be the most expensive destination. Santiago de Compostela might be the second most expensive. It's where the pilgrims end their hikes, so there's a lot of demand there.