Normal oxygen saturation levels depend on altitude. The normal saturation at sea level is 96 to 98%. At 5,280 feet, it is 92 to 94%. So your reading of 96% by arterial blood gas (ABG) is normal. An ABG certainly measures oxygen saturation, but it does not measure lung function. Spirometry is a simple breathing test that measures lung function.
The focus of any further evaluation should be on your headaches and vertigo. This is the most likely cause of your shortness of breath. The best way to know if there is a neurological cause is to be seen by a neurologist.