Why has my 5 year old had a runny nose and cough for three months now?

[ANO111]

My 5 year old daughter developed a runny nose and cough in early September. The mucus from her nose always drains down the back of her throat (postnasal drip) and that's what makes her cough. She's been to the pediatrician twice, and both times the Dr said her lungs were clear. The Dr prescribed amoxicillin on the first visit because her symptoms had been present for 3 weeks by then. They cleared up while she was taking the medicine but came back about a week later after finishing the medicine. There was a refill on it so I started the amox again when her symptoms hit the week and a half mark. They cleared up a little, but I eventually took her back to the pediatrician. This time the Dr prescribed amox-clav. Her symptoms cleared once again while on the antibiotic and returned three days after the antibiotic was finished. The Dr also suggested a nasal spray like Flonase but I wanted to hold off on trying that. I finally gave in and bought some yesterday. Although I'm not sure it's allergy related because the symptoms go away while on antibiotics. I'm starting to really get stressed about not knowing what's wrong and not being able to help her.

Why do her symptoms go away with antibiotics but come back soon after treatment?