I've just dealt with this issue with my own 4-year-old. His original pediatrician brushed me off for *months*, telling me "kids cough" and "give him honey." Such bs for a kid coughing till he vomits every night ....
We saw a pulmonologist who said reactive airway disease with significant nasal allergies. Started Flovent and albuterol, with nasonex. After a month and a second (expensive) visit to the pulmonologist, my kiddo was STILL coughing.
We saw a new pedi and he through there was definitely some post nasal drip going on, but because my kiddo had a chronically runny nose, he said there was probably a secondary , bacterial component going on. So we started Claritin daily and did a round of Suprex (a cephalosporin antibiotic), and he changed his inhaled steroid from Flovent to qvar.
Supposedly Flovent is a bigger molecule and therefore more difficult for little kids to get into the smaller airways, whereas qvar is a small molecule, so it sometimes works better, at the same dose, than Flovent.
3 days into this new regimen, and my kiddo has totally quit the day time coughing, even with activity. And he has only coughed 1-2 times a night. After 6 months of non-stop coughing, it's like a miracle.
So for your kiddo, I'd try Claritin daily. Ask about adding nasonex to your regimen, and ask about switching Flovent to qvar. Then see if a round of antibiotics won't help - adenoids can get infected & drain constantly, making anything else you do moot.
Good luck.
Eta: I use a syringe to give my kiddo meds. It's much easier than fighting w/ a spoon or medicine cup. I also mix most meds into a small amount of Hershey's chocolate sauce, and suck it all up into the syringe and squirt it into his mouth. Some kids do better if you give them the control to administer their own meds, too.