you’d still be in state if you maintain your Texas residency while attending Case Western, so you’d have the same chance of admission to Texas med schools, all else being equal. But yes, you might make connections in state that would help win you admission, if you go to a UT school.
But the real issue is, do you qualify for need-based aid? If not, how can you justify spending an additional 160K on a Case Western undergrad degree vs a UT Austin undergrad degree?