I would suggest exercising at home, maybe in your own room so that you have the privacy to exercise by yourself. Or you can exercise with a workout buddy.
When your running ,walking or jogging even if your dripping in sweat you shouldn't fell embarrassed you should feel proud and make it obvious that you are proud maybe wave a the them ,have confidence and they will know if you do or not so be proud put your head up high.
Well first you have to be confident about yourself and know that you are doing exercises for yourself and not for nobody else.
There's nothing to be embarrassed about exercising. Your 'fatty' friends might laugh at you, but just ignore them. Exercise like nothing has happened and if it's possible, please exercise alone or with your healthy friends.