Have you ever tried a sitzbath in Epsom salts (not too salty of water, just enough to make the water not sting) on the day before you expect your period? (Or two days before, and then one day before?) It might keep the duct or pore open and draining in a natural way.
Also, do you think pads really increase the possibility of it happening? For me, when something irritating happens around the time of my period, I usually forgo tampons altogether in favor of pads. (They do make them without chlorine, etc. , there's Seventh Heaven and other "natural" brands.)
I don't suggest this because tampons are necessarily the cause of cysts and sensitivity, but in general women put in a tampon the same way each time (managing the lips with one hand and inserting the tampon with the other) so there's the possibility that either your fingers or the way you put them in, causes some kind of bacteria to be spread to the spot that 'cysts up' every month. In particular I'd wonder about this if you're left-handed, since you say your left duct usually acts up.
My main thought is, though, that the soaking in warm water is likely to help it stay a pore or an open duct, and not turn into a cyst. The Epsom salts (you could even use some table salt) are mostly to keep the water free of bacteria.
If none of this helps and you've tried it all before, definitely tell your ob/gyn what is happening. I'm not sure antibiotics are a good idea in this situation -- they do kill bacteria (and yay for that) but also kill vaginal flora that is very useful to you. If you're on antibiotics anyway (such as, for your skin), that might even be part of the problem.
Good luck, write back to say whether the sitzbath idea produces results.