If you have an unexpired employment authorization card with six months left, then this may help. In addition, if you are eligible for a STEM extension of your employment card, depending upon your course of study, then you should contact your University's International Student Official (ISO) to see whether the I-20 may be amended, if necessary.
Each State Motor Vehicle Office is supposed to have an office to confirm whether a student is in lawful status. That is, they have access to the SAVE system. If you cannot get proper answers from your ISO, then I recommend that you discuss this matter with the attorney who is assisting you with the H1-b, if any. If not, then schedule an appointment with an experienced immigration attorney, who understands these issues. Perhaps, the attorney can make contact with the Department of Motor Vehicles, where you are having challenges.
The above is general information and does not create an attorney client relationship.