The only state universities that guarantee meeting full-need to out of state students are UVA and UNC - Chapel Hill. The others are extraordinarily stingy because they have to reserve most of their aid for in-state students (whose parents' tax dollars make the college viable). What I would recommend, if you are really intent on going to Ohio and can't pay the full tuition, is that you should look at the merit aid opportunities
. These can really help make Ohio more affordable.
Although I really don't think it's a good idea to go somewhere really expensive for undergrad if you're going law.