What is your budget? Without this information, recommendations are essentially worthless.
That said, Rose Hulman is a fantastic option for you. You might even get close to a full ride. Very underrated university IMO and focused on undergrad teaching and research. Have you looked at Cooper Union in NYC? Another underrated fabulous school. Clarkson in upstate NY is another good option.
Frankly, applying to OOS to public ivy schools is a waste of time and resources. I would not personally do it. Instead, focus on in your budget private schools.
RPI will probably give you “merit” money to bring cost of attendance close to your in-state flagship, but you won’t be happy there. Not strong on humanities, lots of nerdy engineering guys, not enough women, and that’s never a good formula for any gay person, male or female.
Some thoughts on RHIT… some good, some bad for the OP. First, it is a great school, ranked #1 among schools that don’t offer PhDs for almost all engineering disciplines. The school focuses on undergraduate education rather than research, and the professors are very engaged with the students. They cap class sizes at 30 students, compared to a couple hundred at large public universities. They are very generous with merit aid (they offered my a package equal to half the tuition).
Now the bad news… it is in the middle of nowhere, Indiana. It is not even in Terre Haute; it is five miles outside town on Rt 40. As my son framed it, “Rose would be my top choice if it were not in the middle of nowhere.” The student population is heavily skewed towards men (I think only 20-30% are women).
Rose has a summer STEM program where you live on campus for 12 days and do a project called Catapult. It is a great way to learn about the school and see what it has to offer. If interested, contact them right away as the deadline is soon, if it has not already passed.
RHIT (Rose Hulman) may be a fine school but the students are 79% male and off my kid’s list for that reason. I joined a parents’ group and it seems that the students are worked pretty hard.