If you think he has allergies, like if it runs in the family, then in addition to the Flonase, he might also need an antihistamine. Good choices at his age might include Claritin, Clarinex, Zyrtec, or Allegra. A decongestant or a combination antihistamine with a decongestant, such as Claritin D, Allegra D, or Zyrtec D would also work. If those don't work, allergy testing and allergy shots might be helpful.
Chronic sinus infections are another possibility with these symptoms. To help determine if that is the problem, a sinus CT scan might be helpful. Stronger antibiotics or a longer course of antibiotics, like for 4 weeks, might also help to finally stop his symptoms.
If he is missing a lot of school and you aren't getting a clear answer about what is causing his symptoms, a second opinion from a specialist might also be a good idea. Since you have likely already been evaluated by an ENT doctor when he had his tonsils out, you might try and see a Pediatric Allergist this time.