So if I understand correctly, you're fermenting in a conical? What is your secondary, glass carboy? I've had great success dry hopping in the primary and sometimes the secondary, as I usually don't secondary, because, like you I am worried about oxygen. However, everytime you open the lid of the fermenter you are letting oxygen in. I've never had any problems with oxidation, even after opening the lid several times, dry hopping, gravity check, etc...
Are you dry hopping a lager? Since you mentioned a diacetyl rest. Can't you gently pressurize the conical, in essence purging the oxygen out of it. If you're dry hopping in the secondary, you could do like a few brewers I know do. they keep a small co2 tank just for purging their vessels. Dump your hops in, then put the gas in the vessel for a few seconds. Since c02 is somewhat heavier than the atmosphere you should get a nice blanket, protecting you from harmful oxygen.
EDIT: I'm just posing this question from my own experience, not saying this is what's happening to your beer. Anyways, I have two 6 gallon Better Bottles and their dry trap ferm. locks. I noticed when I cold crash in these things, they get all misshapen in the fridge after a couple days. That dry trap keeps it so air tight nothing can get in, however, when I remove it to rack, air rushes into the fermenter. I'm only guessing that since the pressure inside the fermenter changes with temperature, that's why it literally sucked air back into the fermenter! Anyone else experience this? Could cold crashing be adding oxygen to beer, because of the pressure change?