You need to check each website to see if the college awards any kind of aid to international students…and if so, the limitations, and the types of aid awarded.
You are correct as far as public universities go but private ones like Illinois Tech often do give merit aid to international students. Since there is still time, look into other private engineering schools and then try to see which one gives you the best package. If there is a university that gives you a package that is close to affordable, contact the admissions office and see if you can negotiate for a bit more assistance to make it work for you.
Looking at your ACT scores and grades, you should be able to get in to some private engineering schools without trouble and with possibly substantial financial aid. You still have some time and some of these schools even have rolling admissions so you can even apply after the end of January. The list you gave in your original post was all public institutions and those are the least likely to give you any aid. It is really unfortunate that your college counselor at the high school has no idea how to handle a situation like yours. This advice should have been given to you in October.
The requirements for resident tuition for Pennsylvania should be in the link below.
http://professionals.collegeboard.com/testing/international/state
Does anyone know any private engineering schools are sort of well-known?
I will be applying to Illinois Tech soon
Well known depends in what circles. In the Chicago engineering community, Illinois Tech is quite well known. It is less well known on the coasts but our students often get jobs there as well. This is the same with other private engineering schools like Drexel, Stevens, Kettering, and so on.