2 y/o = 1 (one. WON.)
Mom = 0 (zip. zilch. zero)
He is winning this battle, mama! He is PLAYING you like a violin!! He KNOWS how to push your buttons, melt your heart, wrap you around his little pinky and GET HIS WAY!! Enough already. Who is the boss, anyway? Or in this case - who SHOULD BE the boss? If you don't have the heart to say no to something as simple as a binky, I shutter to think what else you could cave in to because of his adorable, angelic face.
You MUST go cold turkey, however that feels right to you. Throw them out, have HIM throw them out, tell him babies elsewhere need them and make a big deal of him 'sending' them away (or giving them to someone you know with a baby) or having the Paci-Fairy come, or whatever other creative thing you come up with that you think your son will understand and embrace.
My son's pediatrician told me recently that at that age (he is 2 also), boys are unsure whether they WANT to be a big boy yet, so don't make it about that. (I called because my boy was not eating much and she said he was asserting control of what goes in his mouth. The only things they have real control over are what goes in, and how/when it comes out, so potty training could be a power-struggle as well - that's why we were on the 'big boy' subject.)
OH! Don't try doing two BIG things (in child's eyes) at once: taking binky, potty training, going from crib to toddler bed, moving, divorce, changing rooms, weaning, have another baby, etc. should all be seperate events with time in between for optimum success.
I do not recommend cutting the binkys - chilldren have been known to choke on them. Sure, it MIGHT NOT happen to your child, but WHY chance it?!
Your biggest obsticle is Dad. How often does he stay there? If he is there several times a week, it could be hard. If he visits every other weekend or so, do it the day he comes back so he has the most time to get used to NOT having it before going back to Dad's house. Then tell Dad when he picks up/you drop off that there are no more binkys.
The sooner the better. The longer you put it off, the harder it is. But you already knew that, didn't you. Best of luck wished for you! (I hope I wasn't too harsh, but it really is time. :))