star,
It doesn't sound like a heart problem to me.
That feeling of not being able to get a complete breath and so you have to beathe deeper and deeper is a psychological problem most of us have felt from time to time. I don't know what causes it and it is vexing.

It's not really what "shortness of breath means," which is really more shallow gasping.
A good test that is totally free is to see if you can run at a decent clip for 5 minutes and see if your arm and chest symptoms worsen. That would point more towards your heart.