Your doctors would be wrong. My twin boys were diagnosed with asthma when they were one. Get a referral to a pulmonologist. One of my boys had the traditional wheezing and the other had severe cough only when he had a cold. BOTH have asthma. The pulmonologist put them on Singular and Zyrtec every day with the nebulizer as needed. Also on an inhaler (cannot remember the name for it but it's similar to Pulmicort).
I feel angry your docs aren't sending you to a lung doc. Mine did the same thing. I had to fight for that referral. We spent the first year of their life in the doctor's office and using the nebulizer at least one cold a month. The doc saying oh it's just a cold. Do this neb treatment and watch your baby wheeze. GRRRR. Going to the pulmonologist changed our lives.
One of my boys is using a nebulizer this week (seasonal allergies) but we're not on Singular or Zyrtec anyone. They have basically grown out of it but my wheezer boy needs the neb when he gets a bad cold. Your son may grow out of it too.
Good luck! It always feels good to know other moms are behind you and believe me we are!