Welcome to HCM.
I think you might be suffering from Acute Dacryocystitis
, which is an inflammation of the lacrimal sac. Since the normal path of the tear film drainage
is blocked, tears are rolling down.
Give hot fomentation twice a day with Oral anti inflammatory drugs two times a day until you get an appointment. Since it has been there for a week, you will require a dose of antibiotic for 3-5 days and ophthalmologist's opinion to see the status of your lacrimal sac .