My application was filed as regular application in early July and got approved after one month. But mine is not Texas center. However, it should not take 5 months to process the application. I believe that you will have good status at least wihtin 6 months after filing. Belive me, 6 months is more than enough.
If the university is willing to pay more, of course you can ask them to change to premium. But if it hires its attorney, I do not think the attorney will suggest change.