This post is very unclear (to me). What do you mean "J2 to H1b papers" ?? You have the application? Or you actually were approved for a change of status? If you were approved, you don't have to do anything except if you leave the country (after which you'd need a new visa stamp). If you are talking about having just the application, and your DS-2019 is going to expire in July, and you are subject to the H-1B cap and all of that, you may have to leave depending on your particular circumstances.