Suggest Treatment For Persistent Dry Cough In A Child

Question : My son has been sick for 7 days. Started off with dry cough. He's asthmatic so we were instructed to do 3 nebulizer treatments a day and neti pot. Finally after 4 days took him back as cough was worse almost as if he chokes and temp low grade. Gave him amoxicillin on Thurs. which he has taken before. He had 3 5mg doses from noon to 8pm. On Friday he had another dose at 10 am. at noon we went for lunch and he had fries and a root beer. he has no allergies and never had a reaction. He eats all nuts etc. After lunch he broke out in hives. Took him to ER. hives subsided with benedryl. Noticed he was having a asthma attack unrelated to hives they think. Cough was continuous like it has been all week with me. Did chest X-ray and all good. Did 26 puffers of ventolin and 8 puffs of steroids and oral steroids. Much better and sent us home. No idea about the rash because it went away. Told me to give him 8 puffers every 4 hours for 48 hours. After we came home hives came back with a vengeance at 8pm. Benadryl took them away then they came back in early morning around 1am and went away on their own. Now hives back again at 10:30am and concentrated on his bottom which they never showed before. They are extremely itchy so I gave Benadryl to him again. I never gave him any more amoxicillin (last dose was Friday morning before lunch). Cough is still bad but better than it was I think the puffers are helping. What are the hives coming from? Is it an allergy? Is there a virus with hive like symptoms? Hives are irregular shaped and start off small then welt and merge with each other. looks whitish in the middle and redish on the outside. Not sure what is causing this