I don’t know what to say, this is obviously a very difficult situation. It’s good to see that you realize you need help. Now, are you searching for help in the right places? I think it would help a lot if you would see a dietician and possibly a psychiatrist.
No, you should not have to give up running as long as you realize where running fits in a healthy lifestyle. My guess (although I’d strongly suggest confirming this with someone who knows more than me) is that you were probably starving yourself when counting calories, which led to the binging when your body needed calories. Counting calories MUST be handled differently when you’re a runner than for non-runners because your body is burning more calories.
Again, I truly hope that you go to get the help you need.