Right now i am in the same situation.. Even being state university it is not e-verified. I did not think about that when I got the offer.. Thus I had to find a new employer to extend my Opt. If you know some indian concultancies they might want to give you a letter indicating you work for them with I983 training plan and get the referral from your DSO...and you might be able to continue to work for the same company.. a tedious process though.. And that consultancies might ask for some money as well...
I advise you take that offer and while working you can find another job with an employer who is e-verified ....
Best.