Your situation sounds very similar to my children’s. Approximately $12,000 per yr is all we can afford per child even though our expected contribution is significantly more than that. They are gifted high performing kids with high levels of academic achievements. We take the position that they can apply to schools where competitive scholarships eixst, but if they are not awarded one, it is no longer an option. We would not let them apply ED to any school where it is blanantly obvious that we cannot afford to pay w/o significant competitive merit.
I would strongly encourage you to apply early to several high merit schools and look into honors programs, so into schools where dept scholarships might be available. Wishful thinking is not going to bridge the gap you will face a lot of disappointment.
Fwiw, my kids have all loved where they have ended up attending even though their academic achievements are fare beyond the typical applicant pool. Plenty of high achieving kids are in the exact same predicament and so lots of peers are found on theses campuses. My kids have had fabulous opportunities at their schools. Don’t think that you are settling. Find out just what you can gain by attending those schools (honors programs, undergrad research, mentoring, taking grad level classes as an UG, etc.)