An independent Texas would have no effect on existing student loans. Student loans are contracts between the lender (federal government) and borrower (student). These contracts will need to be fulfilled and will not be voided. However, the federal government has facilitated predatory lending practices aimed at college students. Texas has taken a strong stance against such practices in other areas so it is reasonable to believe that an independent Texas will not tolerate such activities.
Universities would continue to function as normal, the only difference is they would be classified