Often you are better off applying to private schools. They have bigger endowments and can offer more money than public schools.
Smaller, private, liberal arts colleges are desperate for students. My second daughter got a lot of solicitations from them based on her well-above-average verbal PSAT score, even though her math score was pitiful.
They’re offering $10-15 thousand tuition discounts, sight-unseen. One could probably get free tuition if you just paid the room and board!