Listen! First, you can check the e-verification status of your employer online. As far as I know, most schools are e-verified. Even if it is not e-verified, its free to join for them. Get that job, work for couple months, develop good relationship with your supervisor/coworkers. Later, request them to join e-verification program, explain your situation. There are nice and understanding people out there.
Good luck

Great advice, the only concern is if they are not willing to go forward, I may loose both of these jobs. This job is not great but that is what most post doc do. I hope I am doing good here and can have another year extension and possibility of teaching, if works fine. But the other job is permanent and full time.

You can obtain 17-month extension from current employer and change the job. I am not sure how long the new company will wait for you.

Most of the government offices are not e-verified and will not help you for your H1B. Better check first before making any decisions.

Good Luck.

Thank you again, yes, I looked the specific department on e-verify but they are not listed in USCIS e-verify list. Anyway, they are doing documents verification now and the offer is tentative until it clears. But anyone who is employed in State government can make this more clear. Also, each case are different.