The H1 approve required the Waiver offical approval notice. You can start to apply for H1 with DOS letter, but you must get the final notice before you get your H1. So it is not good for you to use expedite service.
The better thing for you is applying the J1 extension firstly. I hope you have enough time to do it. Ask some lawyer service.
For you J1 waiver approve notice, maybe you can do as follows:
1. Go to your local INS (or bureau) place. They will give your information about what happens of your notice.
2. Call the custom service. I did it 2 years ago. It is very hard to connect it. But you need to try many many times. I cannot sure whether this phone exist or not now.
For my case, I didn't receive my approval notice because I moved to a new address. I left the mail address at my application form is my office's address. So I didn't update my new home address. But they said they must send it to my home address. When the mail returned, they destoryed it at once. After I called them and updated my home address, they send me a duplicated approval notice. It works.
Good luck.