After I read your post again carefully, it is true that your H1 status has a problem as your international office said. The problem is you had not worked for University B with H1 status before you used AP and thus, you were not qualifed to work us ing H1 status for the University B after you used AP to enter US. You should have got the H1 visa based on the University B sponsor and not used AP. So, you may need to go back to China to get the H1 visa to come back as the International office suggested if you want to be H1 holder. But if you do not want to keep H1, you can use EAD to work. Also, because you have the EAD all the time including the period you worked for the Univ. B,, your I-485 will be OK even if you did not show the EAD to the employer (you can show it to the employer now if you plan to use the EAD to work). So, do not worry.
