Cough/mucus in chest. can't expel without stridor/asthma attack. on amox/prednisone 3 days. mucus not better. are more meds/tests needed?

Depends on why: What to do next depends on the reason you are ill. If you have a viral infection, antibiotics won't work; influenza is not prevalent this time of year so looking for it won't help. If you have pneumonia, Prednisone isn't helping, but you might cough for a week after the infection is gone. If you have post nasal drip you need nasal steroids and saline rinses. Go back to your doc and try again.

Asthma attack: You know your problems are related to allergies and asthma and you say you are not using any inhaler you should talk to your doc how to control and relieve asthma symptoms you need to take controller meds like inhaled sterois with/without long term bronchodilaters, like advair and dullera twice a day, singular and reliever mds like albuterol or duoneb. In acute cases you need oral steroids short c.