Firstly, you need to apply J1 waiver before changing any visa, such as H1. J1 waiver is your business, your university cannot help much. But they must write you a letter to support your future work after your J1 expired. This letter is for Chinese embassy only and you can tell them it is not a promise. But if you haven't this letter, it is hard for you to get this J1 waive.
After you get your J1 waive (maybe need half year to 1 year), you can change your visa to H1. Before you get this, you must keep your J1.