Given your history of a surgery for a tear duct obstruction and that it has come back again and is even in the other eye, I suspect that there are other issues at play here than just a possible tear duct obstruction. That is usually not a very common reason to have this tearing. By far dry eye is the most common cause of tearing. It sounds backwards to attribute tearing to a dry eye, but when the eye is dry, the eye sends a signal to the brain that tells the eye to water, but the watering is usually not sufficient to keep the eye hydrated over the long haul. There are many reasons to have a dry eye. Some of the most common are: not making enough of your own tears, having eyelid inflammation called blepharitis, allergies, living in a dry/windy/dusty environment and some medicines (antihistamines commonly).
One of the first things to do to start treating dry eye is to start using artificial tears 4x/day for a few weeks. It must be consistent, daily use, however, or it won't work. Then, if there is mattering or crusting on the eyelashes in the morning and the eyes burn, sting or feel more irritated in the morning than the evening, then a daily regimen of hot compresses on the eyes x 10 minutes, followed by scrubbing the eyelashes with a dilute baby shampoo solution will help treat eyelid inflammation. If these things are done consistently, it is likely that you will have a noticeable improvement, but you must be consistent about them.
Knowing your history a tear duct obstruction, this could also be playing a part in the tearing if there is any residual blockage. The fact that your dry eye responds to steroid tells me that there is likely an inflammatory component to the dry eye, meaning there is likely some blepharitis that needs to be treated, which is what the hot compresses and eyelid scrubs are for. It may be that you need the steroid a little to kick start your dry eye therapy, but I would start with the lid scrubs and compresses done concurrently with the artificial tears for 3-4 weeks before tapering down on the artificial tears.
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I think you are on the right track. I would try the hot compresses and eyelid scrubs with using artificial tears 4x/day for a month and if things aren't improved I would see your ophthalmologist to investigate other causes of dry eyes and/or tearing.
Does that make sense? Any other questions I can help you with?