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We hired someone to work with my D on her common app essay, and paid about $1000. When we finally saw the essay – parents weren’t allowed to until the student and consultant thought it was done – it was clear that the help had been worth it. The essay passed the backpack test (if someone found a backpack on the street with this essay in it and no name attached, would they know who wrote it) in spades; I don’t think another kid on the planet could have written it, it was so specific to her. Some magic was worked in helping my D to summon those thoughts and connections and get them on paper.
The good news is that a lot of the narrowing has been done for you, given the financial parameters. Though your son's test scores make him a contender for the Ivies, super-selective LACs, etc, any school that gives purely or primarily need-based aid and no full-ride merit scholarships isn't going to work, as you saw from the NPCs you ran. It doesn't make any sense to apply to those. There have been good suggestions above about state schools with automatic scholarships like Alabama, and the possibilities open if he makes National Merit.
There are some special-application full-ride scholarships at LACs, though extracurriculars, leadership and essays are important for those. The schools will generally be in the South; here are a couple: