Apply to everywhere you're interested in. You'd be surprised at how much money these schools throw around.

I had fairly similar numbers (a bit weaker?), and I was rejected from Stanford but I did get into an engineering school on the east coast that's quite a bit less street famous but similarly well-regarded in industry.

I was broke as shit when I applied. My family declared bankruptcy during my freshman year. My current school is one of the most expensive in the country - by the time I graduate I'll be $80k in debt after having received $160k in scholarships. I got heavily involved with my school freshman year and when my family's financial situation went especially south I was at risk of dropping out. I wrote a letter to my school and they gave me another 6k/semester.

Now, this might seem absurd, but I'm also into STEM like you. I made a very very good choice taking on this debt and here's why: after my sophomore year, I got an internship in Silicon Valley that pays ~90k/yr. I'm currently back at that company for an extended internship (taking a semester off) for ~98k/year. In a few months going to a different company at ~100,000k/year.

When I graduate, I'll likely have an offer from each of those companies, each being in the ~120k salary + ~100k in stock range. My earning potential only goes up from there.

If you can get into a top tier engineering school, even if it means taking on enormous debt, DO IT. You will have to work hard to make it pay off quickly, but you'd have to really suck for it not to pay off within 10 years.

Savvy Americans are very anti-debt right now. The problem with the attitude is that debt in and of itself isn't bad. It has a specific, and very worthwhile, purpose. This is that purpose. Leverage it.

TLDR: Apply everywhere, be surprised at how much aid is available, if you're going into a highly lucrative field, debt is okay as long as it'll pay off. If you go to a top STEM program, it will pay off.