Can it be that our air will be unbreathable in a few years due to CO2 emission? I think I may suffer from global warming anxiety and I accuse CO2 for my shortness of breath. Is it rational?
No, you can't blame the $CO_2$ concentration per se for your shortness of breath. From the Minnesota (U.S.) Department of Health:
The outdoor concentration of carbon dioxide is about 400 parts per million (ppm) or higher in areas with high traffic or industrial activity. [...] Carbon dioxide concentrations indoors can vary from several hundred ppm to over 1000 ppm in areas with many occupants present for an extended period of time and where outdoor air ventilation is limited. [...]
Carbon dioxide is not generally found at hazardous levels in indoor environments. The [Minnesota Department of Labor and Industry] has set workplace safety standards of 10,000 ppm for an 8-hour period and 30,000 ppm for a 15 minute period.
In a worst case scenario (energy consumption growing and being met 100% with fossil fuels), the U.S. National Oceanographic and Atmospheric Administration projects a peak of 900 ppm in 2100 -- still less than the indoor concentration in many places today.
However, there are several related effects of climate change that you could blame for your shortness of breath:
As far as displacement of oxygen, CO2 is not even close to a concern. Air is already close to 80% nitrogen, and about 20% oxygen. CO2 is only about 0.04% of the atmosphere, so even if CO2 were multiplied by a factor of 10,000, the concentration of oxygen would decrease by only about 4%.
However, the human body relies primarily on the presence of CO2, rather the absence of oxygen, to detect when it's not getting sufficient respiration. As little as one order of magnitude (i.e. multiplying by ten) can be enough for there to be health effects of CO2, but only the most pessimistic projections have us doubling CO2 concentration in the next hundred years.