Are you diagnosed with an Anxiety Disorder, on meds, etc?
I have struggled with severe chronic anxiety and panic disorders for around 4 years now and have been able to get it under control. I really think that you need to treat the disorder itself...therapy, medication (I am on Zoloft and have Klonopin for when I feel an attack coming on), etc. Breathing has been and continues to be an issue for me as well. I panic when I'm falling asleep because I become conscious of my breathing and I start to fear that it will stop while I am asleep...so I definitely can see where your fears come from when your breathing and heart rate are accelerated while exercising. The most that I can say is that excercise has helped my anxiety tremendously, and I really think that you should stick with it. Your fear of a heart attack (something that's a trigger for me, too...your triggers/fears are very familiar to me) is likely based on your discomfort with your weight, and not exercising is not the answer. I feel for you. I know how anxiety can cripple one's life. I highly recommend seeking professional help for your mental health so that you can focus on working on physical exercise for your health. Unfortunately, as I'm sure you are well aware, no amount of rationality from other people will convince you that you are, indeed, safe to do physical activity and that you are not/will not be having a heart attack...even though you know deep down that they are right.

Hang in there. Get some help for the anxiety so that you can focus on your physical health. If you have any questions or need to talk, feel free to PM me!