First, you need to build your list from the bottom up.
That means finding a safety plan first: Maybe you go to community college for two years to save money, then spend the investment money over two years instead of four to go to a private school. Maybe you find an in-state tuition 4 year option that you can afford. If you qualify for any of the automatic full tuition awards linked above and you would actually attend if admitted, you can put those in your safety plan. But, don’t get in the situation where “I was admitted full-tuition to Far Away University but I don’t really want to go there.” That is not a safety plan. Look at what the schools actually offer you besides just a cheap price! You have to want to go there.
Merit aid awards are typically going to come in with prices above your in-state public options. That’s just the nature of the game: half-tuition or less awards are most typical. Many Net Price Calculators will estimate merit aid if you enter your test scores, GPA, and number of AP classes. Check the schools already on your list. The merit award offers will be at schools where you are among the top applicants, not at schools where you are one of the great middle. You need to be at the very top of the applicant pool to get more than half-tuition, and you have to be low enough in the rankings that the school actually offers awards that large.
Here are some schools to try: http://diycollegerankings.com/50-50-highlights-colleges-with-most-generous-financial-aid/5403/
Another is to look at Kiplinger’s list of Best Value Colleges. (This list mixes need-based and merit-based aid, so check the Net Price Calculator to be sure a school works in your situation.)
Here are a few other tips: http://www.mykidscollegechoice.com/2016/02/13/top-8-ways-to-find-affordable-colleges/