I did the stretch between Leon and Santiago at the same time last year. Most days I finished walking by 2 or 3pm and never had any issues finding a bed. The leg between Sarria and Santiago is notoriously busy, so on this section you might want to arrive at albergues earlier rather than later. The only occasion I was unable to find an albergue bed was the second last night on camino and it was after arriving in town well after 5pm. I did find a twin room in a pension for 40 euros though, which I was able to share with a friend.
Re. weather, I was lucky and didn't really have any rain between Leon and Santiago. Generally speaking I found the mornings were cool so I would wear my rain jacket, which didn't provide warmth, but did keep the cool air off. By 10am generally it was warm and by midday it was hot. I am speaking generally and there were some days it didn't get too hot. What constitutes warm and hot varies person to person though, just as the weather varies year to year of course. In any case that was my experience of the same period last year.