A cough with yellow-green phlegm and chest congestion after a viral infection usually means some mucus is still settled in the airways, which is common and not a new infection on its own. Your asthma and the recent weather shift make the airways more reactive right now, which explains the discomfort with deep breaths. Normal breathing and no fever are both reassuring.
Next Steps
Steam inhalation, warm fluids, and your usual asthma inhaler as prescribed should help loosen this over the next few days. If the phlegm color turns darker, or breathing feels harder, get it checked.
Health Tips
See a physician or pulmonologist if this doesn't start clearing within a week, since lingering phlegm after a viral illness sometimes needs a proper course of treatment.