Being anxious about people laughing at me when I exercise.

What you are experiencing resembles a couple things.  First of all, this sounds like a Social Anxiety Disorder, which is sometimes called Social Phobia.  This is characterized by a fear that you are being negatively judged by other people.  These fears can be triggered by perceived or actual scrutiny from others.  You feel humiliated and embarrassed about your weight, to the point of thinking everyone must be laughing at you.  That apparently only happens in a specific situation, when you are exercising in front of other people.  If it happened at other times and in other situations, it might be considered a generalized social anxiety disorder.

It is also somewhat similar to Body Dysmorphic Disorder, in which a person is preoccupied with an "imagined" defect in appearance.  However, you are uncomfortable about actually being overweight. Then the question is whether or not your preoccupation and concern is "markedly excessive."  If that concern adversely affected your eating habits, it might be the reason for Anorexia or Bulimia.

In any case, you have pretty extreme anxiety about exercising in public, and this is causing you a lot of distress, including panic symptoms like a racing heart beat.  To keep that in check, you avoid the situation entirely, therefore not getting the exercise that might help you lose the weight itself.  So it is kind of a vicious circle, that has physical health consequences as well as mental health ones.

You say that you don't exercise in public.  Do you exercise in private?  Granted, the equipment at a gym can be helpful, and you don't get the benefit of that, either in the specific kind of exercise you can do, or being better able to establish an exercise routine.

You say that you are shy about your body in normal times.  So that is a general concern.  But something about exercising triggers this more intense anxiety.  Does that happen with any kind of exercise in public?  Or is it only when you are doing certain kinds, or on certain pieces of equipment?  If you feel particularly awkward (and laughable?) doing specific exercises, perhaps a trainer or other bodywork professional could help you find alternate ways to get similar benefits.  I think right off the bat about water aerobics, which is excellent all around exercise, with the water both increasing the resistance, and supporting your body as you do it.  At the very least, most of you would be under water, and therefore less able to be scrutinized.  I see women (mostly) of all ages and body types in those classes, and they seem to be enjoying it.  It would help if you could find an instructor who understood various physical challenges, and could adapt routines to fit the individual.

The usual mental health treatments for social anxiety disorder are various antidepressant medications and/or cognitive behavior therapy.  You could ask your doctor about the meds.  And you could work with a therapist, learning how to identify, monitor, and replace your negative and self-defeating thoughts with ones that are more positive, realistic, or at least more neutral.  You can get in the habit of doing that as soon as you notice yourself becoming anxious.  Being aware of, and challenging, your thinking at those times can at least short-circuit the escalation, and reduce the intensity of your anxiety.  And that can help you get on with the exercise, knowing that you are taking care of your health, even though you may still feel self-conscious.

One of the great things about the gym or the spa is that, if you show up regularly, people will respect you for it, especially if you are out of shape to begin with.  They will think it is way better than sitting around at home, vegging out in front of the TV, eating junk food.  I'm not saying you do that, just that it is one common stereotype about overweight people.

Rather than being pitied or scorned for doing nothing to improve my appearance, I'd much rather be admired for making the effort, whether I look amusing doing it or not.  Chances are I'd smile right back at them, and then pat myself on the back ... okay, the shoulder ... giving myself credit for being there at all.  And being able to laugh at myself ... with affection, not derision ... has proven to be one of my most important coping and survival skills, through some pretty rough times.  Perhaps that would help you feel better about yourself, as well.  Good luck, and enjoy!