Hi Sarah!
I see you've already been given some great advice. I just wanted to say that I can 100% relate to the binge-eating. I did a stupid stupid stupid diet that lead me to eating uncontrollably for almost 3 years. To this day I still struggle, but I learned something that maybe can help you.
You need to figure out what's triggering you to binge. I know, easier said than done. But what I learned about myself is that if I try too hard to just eat all healthy foods, that'll last for two days (max) and then I'll binge in unhealthy foods. Trying to hard to eat healthy is what triggered me to binge! It took two years for me to fully realize that. I kind of knew it all along but I so badly wanted to eat healthy everyday. It took almost an additional year to teach myself how to just have a tiny portion of a splurge food a day without triggering a huge binge-eating session. I was so used to associating unhealthy foods with binge-eating sessions that I had to recondition myself to not associate a small splurge with over-eating. It was tough, but now I've learned that when I'm craving something bad (I'm a sucker for lattes), I just have it, and I don't always binge now. Otherwise, if I don't have it when I crave it, I'll over-eat the next day. With this process I've gone down 30 lbs. Now, slowly, I'm trying to limit the days I give-in to my craving. Baby steps. Before, everyday I was indulging in a latte (and maybe even a dessert with that), now I've been having small splurges every other day. It's what works for me because I learned my trigger.
Next time you feel like binging, try and think of what's causing the urge. Is it due to being upset because your family is not overweight (hugs to you in regards to that, btw...all my friends are thinner than me, so I understand what you're going through<3)? Or is it because of something else? You are uniquely yourself and not your family--no one can be you, and everyone has their own struggles. Don't mind them. They may not understand what you're going through, but thousands of other people do (including everyone on this site!).
As far as excercising, 15 minutes is better than 0 minutes! Try and increase that amount slowly. Do 15 minutes, then slowly increase to 17, then 20, 25, etc. Baby steps.
You can do this, Sarah!
