How can I have an oxygen level of 100% at rest and still be short of breath?

They're different. Oxygen levels have little to do with the sensation of shortness of breath. The work of breathing determines how short of breath you feel. When oxygen levels get too low shortness of breath may occur, but many people feel shortness of breath with totally normal oxygen levels. Even with low oxygen levels you may not feel short of breath at rest.
Non specific symptom. Shortness of breath is a very non specific symptom. This sensation can be caused by diseases which affect the work of breathing but not affect the level of oxygen in the blood. For example muscle diseases that make the breathing muscles weak may behave this way. There are a large variety of other diseases that can as well. In this situation a complete medical evaluation is needed.
You should not. You may have to check with your doctor, looking for for any other problems such as the heart etc.
Increased usage. With activity your metabolism increases and therefore you utilization of oxygen. If you extract more oxygen than is replaced your saturation will decline. This can lead to a sense of air hunger, making you feel short of breath. There are other reasons for shortness of breath as well such as asthma, emphysema, congestive heart failure, altitude sickness, etc.

Related Questions

If parts of the lung became hypoxic but the rest of the body had enough oxygen, would this still cause a short of breath feeling?

Theoretically... Consider ventilation-perfusion mismatches classically shown by V/Q scans. To deliver oxygen & release carbon dioxide, you need circulating capillaries & open alveoli. "Parts of the lung [becoming] hypoxic" is not exactly the right phrasing. But, along those lines, if you had a lobe collapse, an embolus, or pulmonary edema, if large enough, they will impact ventilation & cause shortness of breath. Read more...