Can chronic sinusitis cause throat swelling sensation and fullness and shortness of breath?

Not necessarily: Chronic rhinosinusitis can lead to an asthma flare, and a sense of sinus fullness. However, if you are having throat swelling, it can be due to angioedema or to a severe allergic reaction. You should see an allergist as soon as possible. If it happens again, go to your emergency room - this can be life threatening.

Yes: Chronic sinusitis is an inflammatory condition that results in the secretion of lots of chemicals that cause and increase inflammation. These chemicals often spill out of the sinuses and drain down the back of the throat, and can cause the sensations you describe. Topical treatments such as mouthwashes can make it feel better, but you need to see your doctor to get the chronic sinusitis addressed.