NUS OSA usually reserves a lot of rooms for exchange students, more than enough. So they will be giving out rooms before semester or weeks after semester begins.

If you really want to stay, there is one way our batch used to do, apply for holiday stay first to ensure u have a room during July and early August, then towards the end of holiday go to OSA everyday and talk to their admin staff (usually Ms. Khoo, who is in charge of student housing). The lady is quite tough. However, do not give up and you will finally get a room