A very close school friend of mine has this. It's called epiphora if you tear up excessively.
Has your Dr performed an examination of the duct? (They use a probe to check for blockages)
Yes, please go to an allergist for allergy tests to see if you have any allergies that could be triggering the watery eyes. They may be able to treat you.
And if you are in pain, like someone I met was who had their eyes watering from painful headaches, visit an experienced neurologist to try to get some help. I might visit them anyway, since there seems to be a directional component (downward head) to your eye watering.