The reason no one in your school goes to a “t-20” school is probably not because of the school, it’s because of affordability.
If you’re low income, then an out of state prestigious school will most likely be out of your price range. If you’re in Texas, UT-Austin is already a “T-20” engineering school. So is Texas A&M. Those schools are going to be far more affordable than any other schools you could go to. UT-Austin, for instance waives tuition and fees for low income. With your parent’s income what it is, you can pretty much expect free tuition.
Also, check out potential full ride scholarships. You’re a prime candidate for a full ride at UT-Dallas. You could also check out U of H, Texas Tech, etc.