Just was diagnosed with small airway disease / asthma. Prescribed an inhaler, which I used frequently, and now prescribed advair. Is that normal?

Yes, very common. Asthma is treated in a step wise fashion. If the rescue inhaler isn't working well to control your symptoms, then an inhaled steroid would be prescribed, then an inhaled steroid/LABA like advair. Once controlled well, you might be able to de-escalate therapy. I would discuss this with your pulmonologist.
Asthma. If asthma symptoms require frequent use of rescue therapy like albuterol use, causes night time awakenings from cough, frequent er visits for asthma, then one needs an every day preventative inhaler.