Hi, the possible causes of cough with sputum are Gastroesophageal reflux, Laryngopharyngeal reflux, Respiratory tract infection, chronic bronchitis, Bronchiectasis etc.
You doctor may be suspecting acid reflux. That is why he prescribed omeprazole. But you seem to have eiether chornic bronchitis or bronchial asthma secondary to allergy esp. in the early morning.
You need to undergo lungs radiograph to rule out chronic bronchitis, Spirometry for lung function tests. You may be allergic pollens which may be released in the morning. In such cases try keeping your windows closed at night. You can go for centrally acting cough suppressants, mucolytics and steam inhalation after consultation from physician, as well as appropriate therapy. Take care and regards.
Maybe it could be the blankets you are sleeping with? It could even be that there is something else that the doctor is not catching. Take a hot shower with steam to clear your nose and chest before you go to sleep. Maybe it could be your tonsiles? You could go to a ears, nose and throat specialist and ask about that. That same proplem happened when I had an infection on my tonsiles.