In order to keep your legal status here, you have three options below.
Option 1: You have to submit to H1 application and get the receipt of it by June 30; otherwise you'll be at risk.
Option 2: You have to change of your status to B1 other nonimmigrant status or continue your F1 by June 30, otherwise you are in danger.
Option 3: Leave U.S. by June 30 to aviod your "Out of Status", meanwhile ask your new employer to apply for H1 for you and try to get H1 visa in China for return.
You do not have so much time left and the clock is ticking, so make a wise decision for your best.