Do not be surprised if some colleges want your financial information: FAQ: Divorced parents, financial aid, and net price calculators
Regarding your title question, probably about 70% of colleges in the US say that they are need-blind for admission, while the rest may consider financial need in admission. Need-awareness may more likely to come into play for applicants closer to the margin of admission or denial, and larger need may have more effect than smaller need. Need-blind or need-aware status does not indicate whether the college will offer good financial aid for any given financial situation.
However, note that correlates of need or lack thereof are often used in admission (e.g. legacy correlates to lower need, and first-generation-to-college correlates to higher need, even though there are some applicants who are exceptions to these correlations).
Also note that, for colleges where the applicant appears “overqualified”, some colleges consider “level of applicant’s interest” to try to determine if the college is really one of the applicant’s high choices, or a low choice that the applicant is unlikely to attend if admission (because the applicant will have admission to more selective colleges that most consider more desirable). In the latter case, the college may deny or waitlist the applicant in order to have higher yield of those whom it does admit.