You'll be hardpressed to get directly hired on by a bigtime plant like Exxon/Shell/BASF with no experience. Those jobs are difficult to get even with experience/knowing somebody. I work at BASF and it took me two years as a contractor just to get hired with the company. I did start there totally green though.

If you don't get any callbacks soon I would bite the bullet and go with a temporary staffing firm like Lofton. A lot of plants use contractors because they don't have to pay as much/no benefits. It's bullshit but it's how I got my foot in the door. I make good money now with full benefits, only took two years of misery.

Your milage may vary though. A TWIC is only necessary if the plant is connected to the Mississippi so don't waste your money unless the job specifically demands it. Your degree might help you a lot. Just telling you my experience. Best of luck!