Child Coughing, Sneezing After Swimming. On Medication For Suspected Dust Allergy. Could It Be Exercise Induced Asthma?

Hi..My son is 10 year old. He is being trained for swimming competitions. About 2 months ago, he started coughing and sneezing. Doctor checked for allergies and he has 2719.1 Total IgE ans x-ray shows enlarged adenoids. Doctor said it could be dust allergy because there are construction going on around our house and put him on Singular 5 mg once a day, Budecort 200mg (twice a day for 2 weeks and then 1 a day) and nasonex once a day. Although he is better, it did not fully cure him. Even now, once he comes out of his swim training, he has nose blocks for sometime. We are thinking that he is having exercise induced asthma. Do we need to continue this medication for long and advise if there is any other medication that helps with exercise induced asthma.