My understanding is that, if you have low tear production, the plug is supposed to help keep what little moisture you have on the surface of your eye for longer, as opposed to having it just drain away. Then you can rely less on artificial tears. I think that for a lot of people dry eye is a little more complicated than low tear production, though.
For example, I have problems with the oil glands in my eyelids not producing enough oil a lot of the time, which causes the moisture to evaporate more quickly, coupled with low tear production from inflammation they think is probably stemming from an autoimmune response. I really hope this isn't the case, but I think the reason I'm not noticing much of a difference is that I just don't have enough moisture for the plug to trap.
So, anyway, although some doctors don't seem to want to take the time to really nail down what's going on and would rather take a one-size-fits-all approach with people where they just push Restasis on you and get you out the door, there seem to be different causes of dry eye for different people. And, from what I've heard, it seems like the results people get with the tear duct plugs are sort of hit-or-miss because of this.
As far as the lump on your eyelid, was your doctor sure it was a blocked tear duct in the corner of your eye? I just wondered if maybe it could be a blocked oil gland that is really closeby the tear duct. If that is the case, I get those sometimes and warm compresses help open them up and go away, no med's or ointments required. It could be worth a shot to try compresses twice or so a day (just make sure they are warm, but not super-hot) for about 5-10 minutes each time for a few days and see if that helps. Good luck!