Welcome in H.C.M.
I have gone through your query regarding prolong cough and cold along with earache.
As you said that you are suffering from cold and cough for last 4 weeks and you feel severe congestion and headache in morning hours, and as time passes congestion gradually passes but the body pain
Your all above features are indicating that you are most likely suffering from sinusitis
and as it had became chronic thus post nasal drip
had settled and in this way infection spreads up to throat. From throat infection
progressed even ahead to ear via eustachian tube. In this way your ear had started popping and became painful.
Now it will be better to consult any ENT doctor for thorough examination and treatment to avoid complication.
Till you are waiting for specialist you may take-
1- Antibiotic (preferably broad spectrum).
2- Steam inhalation.
4- Tab Paracetamol
I think above answers your query.