Speficially to answer your question about the campuses (Binghamton and Syracuse), for MD applicants and people applying for matriculation right away, they usually tell you at the interview to select a preference right after your interview day (I think it's through MyUpstate, but don't quote me). The options for campuses are typically Binghamton or Syracuse ONLY or Binghamton or Syracuse Preferred, meaning you would be flexible on which campus you are willing to attend. I'm not sure how this works for BSMD.
If I remember correctly about 1/4 of the class goes to Binghamton for third year, and typically its based on your preferences you indicated after your interview or based on when you are accepted (If you are accepted later and select that you would prefer one place over the other, you may not get that preference). I think the best way to decide would be to talk to people that have gone to each campus for third year, and I'm sure if you reached out to admissions, they could put you in contact with people. I'll be starting third year in May, and then I may be able to answer your question with more experience-based knowledge when I start.
In general (from what I've heard): In Syracuse you tend to spend more time on rotation and learn more in person, but in Binghamton you have more time to study outside of the clinic, so shelf scores tend to be very similar. At Syracuse there are a lot more medical trainees (residents and fellows) whereas in Binghamton there are definitely fewer. This means at Binghamton you would have less interaction with other trainees further along in the process, but you probably will get to know the attendings better. At Binghamton you definitely need a car or some reliable transportation (Uber/Lyft or carpool if you don't have your own car), but I've heard it's probably easier to go to Syracuse if you don't have a car because you can walk most of the time. With everything moving virtual now, a lot of research opportunities have been made virtual, however, if you are really into research you would probably have more in-person opportunities at Syracuse. Also I recently found out that if you are in Binghamton and want to do an elective that is only offered in Syracuse, you can stay in Geneva for the duration of your elective so you don't have to pay double rent.