If the DSO in your current school's ISO says no grace period, then no grace period. Only when you graduate, 60 days grace period becomes entitled, if anything other than graduation, it's not entitled and will be any number of days (or zero days) ISO(current or new school) tells you. The next school's ISO will determine whether your need reinstatement or reentry with new F1 visa after they look at your SEVIS record. The longer you get new I-20, the worse.