I have a constant tickle in my throat. I think my doctor is just as perplexed as I am on it. I get it about every 3 or 4 weeks. I am trying to do research on it to decide if there are any specific tests we can run. All tonsils contain crypts.
Crypts are pits in the surface of the tonsil that are like deep narrow caves.
Sometimes the infection is not at the tonsil surface where the doctor's swab samples the bacteria in your throat for culture, sometimes the infection is at the tonsil core- where the swab cannot reach. Chronic strep throat does tend to have symptoms like you describe and these symptoms do tend to get better with antibiotcs - but come back again if the infection at the core remains. I do not think that the white spots are related to your asthma medicines, from your description. I do think that people will asthma can have a hard time with their breathing when they have a chronic infection in the upper airways- including the back of the throat. Perhaps you should speak to your internist about referring you to, or having a consultation with, an otolaryngologist or an infectious disease expert.