If your main symptom is cough you need detailed clinical examination starting from nose, nasopharynx, throat and last but not the least chest.
Then a clinical can comment where is the exact source.
Its also important to know history like fever, appetite, breathing difficulty, nose block etc.
Next Steps
If your cough is not severe, and you have no other symptom you ...you have the feeling that some liquid dripping down your throat from nose, you can try home remedies.
Because of COVID 19 it is not advisable to visit any hospital unless absolutely necessary.
Health Tips
You can take steam inhalation 10 minutes in the morning and 10 minutes at night
And take an antiallergic medicine ( like levocetrizine 5 mg) for a week.