You can stay beyond the expiration of your I-94 if the H-1b visa petition was timely filed. However, even when the H-1B visa petition is approved, it will not be approved as a change of status since your status will terminate at the end of July and the H-1B can only become valid on October 1, 2012. Therefore, there is a gap and you will need to go for consular processing in your home country for the H-1B visa anyway.
Answered on Jun 26th, 2012 at 7:02 PM