My son has chronic sinusitis & asthma. He's on amox-clav 4.5 q2d and takes flovent 2 puffs twice & singular. He's still coughing and breathing rapidly.

Allergy evaluation. Taking antibiotics every other day helps prevent recurrent infections, but the most common cause of chronic sinusitis is inflammation, most often from allergies. Antibiotics do nothing for this. Persistent symptoms despite good medications such as Flovent and singulair (montelukast) suggest that there is an important piece of the puzzle missing. Take him to a board-certified allergist for further evaluation.
Needs further Rx. Chrronic sinusitis and asthma are serious problems at any age. I presume he is being followed by a pediatrician, and possibly also a pulmonoloist (lung specialist), and perhaps an ENT (ear, nose throat and sinuses) doc. The goal is to keep th patient as wheeze free as possible as long as possible. If he is still breathing rapidly and coughing (cough=wheeze), he needs urgent treatment. Take to er.
Doctor Now. If he has asthma I would hope you have some form of albuterol to give him or already have. However by your description he needs to be seen now because you are describing respiatory distress. Go see your doctor.
Give albuterol. Persistent coughing may be a symptom of either sinusitis or uncontrolled asthma but rapid breathing is a symptom of asthma. His Flovent & singulair (montelukast) are not adequate to control asthma at this time. Administer albuterol and depending on time of day either call the physician who manages his asthma or take your son to the nearest urgent care facility or emergency room.
See a doc asap. Sounds like his asthma is poorly controlled. He needs to be seen by his regular physician soon and if he's in distress he need to go to the er. A referral to an allergist or pulmonary doc may be a good idea once he's stable.
Further Rx and Eval. If your son has both asthma and chronic sinusitis, he needs to be evaluated for the "cause." more than 80% of children with asthma are allergic. Allergy can be treated and markedly improve the asthma. I would also want to be sure he does not have cystic fibrosis or Aspirin sensitivity (samter's triad). So he needs further treatment for the flare up and then allergy & pulmonary consultations.