Upper respiratory infections can often outlast a course of antibiotics. My wife had one that lasted maybe 10 years. they discovered this when they did maxillary surgery and found the infection had been eating away at her maxilla and sinus area bones. She had been through many courses of antibiotics. It took the surgeons 45 minutes to remove and cauterize all of the infected tissue.

So, yes, an upper respiratory infection can last a long time.

I am of the belief that flu shots are actually damaging to the immune system. Only live virus vaccines should be used, not dead virus vaccines like most flu shots. The nasal flu vaccine uses a live but weakened flu virus. The Flu shot uses dead viruses.

Have you tried flushing your sinuses with a Neti-Pot? My wife swears by them when she has a resistant cold or such. Use the Neti-Pot with a good saline solution.