Big merit scholarships may be easier to get at less selective schools.
However, need-based aid tends to be more generous at highly selective schools (usually, without regard to state residency).
Over 60 colleges claim to meet 100% of demonstrated need (although their definitions of “need” may vary).
Among them, the ones with overall admit rates > 20% include:
Bates
Boston College
Bryn Mawr (women only)
Colby
College of the Holy Cross
Connecticut College
Davidson
Grinnell
Kenyon
Lafayette
Macalester
Mt. Holyoke (women only)
Oberlin
Scripps (women only)
Smith (women only)
Thomas Aquinas
Trinity College
Union College
UNC-CH
U Richmond
UVa
Vassar
Wake Forest
Wesleyan U.
Run the online net price calculators for any that interest you.
In addition to the above “full need” colleges, you might want to check out some of the USNWR “regional” universities.
Example: Loyola University Maryland, which has a 66% admit rate, claims to meet 90% of demonstrated need on average, and (FWIW) ranks #4 among the USNWR “Regional Universities North” schools.