I think your backstory is very compelling, and you should share it because it is so intrinsic to your personality and who you are. You have overcome a lot and for that you should be proud. Financially, though, you are better off finishing where you are (if you’re at a four year school) and then applying to grad school at Northwestern or Brown. Transfer students have tougher admission odds to begin with (especially in STEM majors), and tend to be at the back of the line when it comes to FA.
Did you research Northwestern? It is a fantastic school but the co-op program can make it difficult (not impossible) to graduate in four years. Given your history of starts and stops, coupled with being a transfer and your need for aid, that may not be the best undergrad match for you. For graduate school, however, you’re building a very strong undergraduate record and you’ll be working with a fresh slate.