ALso, OP, some of the top programs in engineering and the sciences are at large public universities. For mechanical engineering, for example, MIT and Stanford are top - but so are Berkeley, Michigan, Georgia Tech, University of Maryland, several UC campuses, CU-Boulder, University of Minnesota, UT-Austin, Texas A&M, UIUC, University of Wisconsin, Ohio State, Penn State, Virginia Tech, Mississippi State…
Arizona State is actually top-ranked in most engineering areas, and University of Arizona is solid and well-known in most of them too. The Fulton Schools (Arizona State’s engineering schools) offer a full suite of career services even as an alumni after you graduate. You can get a very good engineering education at both schools and end up with a great job afterwards.
But I agree - don’t rule out private schools yet. Most of them offer excellent financial aid for exceptional students. BUT your state’s public universities are also excellent!