You're allowed to be self-pitying :)
Have you reached out to your gp? I know from experience the difference it can make when you have someone supporting you, in your corner and no one else's. If you haven't, please do so.
The world will not fall in because you've messed up a Christmas cards, just forget about this year!
Sending money to your nieces and nephews can be done anytime, cut yourself a little slack. And honestly? If its a bit late it won't matter. Explain to either them or their parents that you're overwhelmed and overwrought. Their parents ought to understand from experience, and if the children sulk then it won't do them any harm to go without for a bit :)
It sounds as though you take a lot of blame and responsibility upon yourself and consequently beat yourself up when you have no need to.
Today - I made errors
Is this how you judge every day?
So you got a bit muddled - but you were dealing with 2 fractious infants, Christmas stress (drives all of us nuts) and we're no doubt sleep deprived. Would you have managed to keep every detail straight at work with such demands on you?? I wouldn't!
No one expects you to be superwoman you know! All you need to do is keep the little ones safe, fed and warm. The rest of your responsibility is to yourself- time to look after No. 1.