You took on something you ended up being not up to doing. I think that while under normal circumstances, if you commit to something, you should do it, you also have to look after yourself sometimes. Your health is being affected and you need to cut back on the load on your shoulders to a manageable level. While everyone's participation in the Christmas Card exchange is important for its success, I'm looking at it thinking, "Nobody's going to die, fall ill, or be seriously inconvenienced by this." It's not as though you failed to show up for a group of carpoolers, or left a bunch of children standing out in the rain after dark with nobody looking out for them, or showed up two hours late for a volunteer position at an event that needed you to run smoothly. I don't think anyone is going to be seriously incovenienced by not getting a card.
Don't get me wrong, I am not saying people should sign up for these things willy-nilly and not follow through, but the OP really does sound as though she needs to prioritize here, and put herself first. And those of us who have not been through a major depression cannot really fathom what it's like. I'm told it can be next to impossible to motivate yourself to get out of bed, let alone write out a bunch of cards.
So I'm going to vote for sending the group one message saying, "Thanks so much for the cards, and my apologies - due to circumstances beyond my control, I was unable to follow through on my commitment to send out cards to the list. I realize each person's contribution is integral to the success of the card exchange, and I hope you'll forgive me. Have a wonderful 2013, everyone."
It wouldn't be a lie - nobody "wants" to suffer from depression and it's therefore a circumstance beyond your control.