There are several conditions that can cause a teary eye. I'd start by seeing an oculoplastic surgeon (or even a general ophthalmologist) to get a diagnosis. If the problem turns out to be a blocked tear duct and dcr surgery is recommended, it can be performed be either an oculoplastic or ENT surgeon. There are significant advantages for the patient if the surgery is performed by a doctor who is experienced using an endoscope.
There are so many things that can cause an eye to tear that you should start with a general ophthalmologist. This is especially true if both eyes water.