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.