I assume your question about a PET Scan refers to determining if the nodules are benign or malignant. I don’t believe a PET Scan is warranted at this time. Such small spots are most likely to be benign and a demonstrable increase in size within 3 weeks strongly favors an infectious or inflammatory cause of the spots, rather than their being tumors. I agree with the pulmonologist that your history does not favor consideration of malignancy and with the recommendation of another CT scan next month.
A serious question is why a previously healthy, non smoking young woman with no prior history of lung disease, especially no history of asthma, would develop acute bronchitis with persistence of cough for several months. Granted it could be asthma but other considerations would include hypersensitivity pneumonitis, inflammatory lung disease related or unrelated to autoimmune disease and recurrent aspiration bronchitis, secondary to gastroesophageal reflux disease. Of course “constant sinus drainage” (presumably a new symptom for you) could be the cause of your cough in which case assessment of your nose and sinuses for infectious or inflammatory disease would be indicated. Given that your problem, with the exception of the nodules, has been localized to the bronchi, bronchoscopy with examination of bronchial washings might reveal unconsidered bronchial disease.
In the meantime, your sputum should be carefully examined and cultured for bacteria and fungi.
Good luck
Have you been diagnosed with COPD? Chronic bronchitis is a component of COPD as well as asthma. I have COPD and I can tell you, you get colds and infections much easier. Advair will cause thrush. Rinse well and eatting yogurt can help. There is no cure for COPD, but with proper meds, nutrition, and exercise. Has your Alphas been checked, there is a gene that can cause COPD even if you haven't smoked.