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I'm happy that you have asked me to assist you with your child and hope that you can benefit from my many years of helping parents like you.
I can understand your frustration with such a long course of coughing. The question neither you nor I can answer is what her lungs sound like vs. how much of what you are hearing is upper airway noise due to drainage and/or bronchitis. I see lots of kids whose parents are sure their child is wheezing when actually they have a lot of noise that transmits into the chest which you can both hear and feel.
The reason I have doubts about wheezing is that she has not responded to albuterol and budesonide, which usually will help true small airway disease causing wheezing.
Amoxicillin is rarely going to help in this type of illness and zithromax is appropriate for some bronchitis (Mycoplasma) but with this long history of congestion I suspect she has bronchitis and may need something a bit stronger, such as Omnicef or Suprax and perhaps even an oral steroid to help knock out the inflammatory reaction going on in her airway.
Oral steroids, just for 5-7 days will also help if she has wheezing, asthma, etc but again, I suspect she has bronchitis.
There have been several articles published in pediatric journals recently about the failure to diagnose chronic bronchitis in children, all of whom have a course similar to hers.
Regarding her doctor, yes and I see this all the time, especially with inexperienced pediatricians, family docs, and ER docs since I work in pediatric urgent care now and we see other doctor's patients who come to us because they want another opinion.
I would ask for an oxygen saturation (pulse oximetry) to be done (a sensor on her foot or hand) and it might be worthwhile to do a chest xray just to be sure she doesn't have anything going on in her lungs (pneumonia, foreign body or object she inhaled or swallowed).
If she has only upper airway noise then she has bronchitis or sinusitis/bronchitis and needs a stronger antibiotic as well as perhaps an oral steroid.
Don't ever apologize for being a concerned parent and if your doctor's ego is too big or feel threatened to listen to your questions then you need another pediatrician for sure!
You are quite welcome and good luck.