Thank you, 8-teacher.
Right now, the offer is email. It seems that things are heading towards good direction. I am waiting for the official offer. The American girl who got the email offer together with me is very relaxed and told me the process with them is slow.
I have asked international officers. They told me that volunteer work at future hiring institution is not allowable. However, consultation at individual level is ok as long as it doesn't have any money.
I also asked a Chinese working at school district (this offer is from a school district) and he told me that if he was the hiring manager, he prefers the J-->H option. He is a director of an office.
Thanks again.