I would have thought CBT techniques are the best thing for this.
It sounds like you need to regain perspective and also accept your own level of distress.
Acceptance First
E.g. It's normal to feel so sad about this. I really upset someone I love and it was my fault. It always feels horrible to be in the wrong. Anyone would feel deeply upset in this situation.
Normalisation
Everyone makes mistakes at some point. No one gets through life without making one or two big mistakes and this was one of mine. It was normal to make a mistake.
Rationalisation
I apologised. This was the right thing to do. I'm glad I did. She accepted the apology. It was the grown up thing to do. I acted appropriately as soon as I realised my mistake.
Nurture.
How can I take care of myself while I feel so raw? Maybe I could spend time with close friends or family who I haven't upset, to remind myself that most of the time, I get on fine with people I love.
Maybe I could watch a few comedies back to back with the children.
Blessings Counting
(I know people think it's cheesy, but it works)
Make a long list of all the things you have done right recently, and then as far back as you can remember. And make another list of all the things in your life, however tiny, that are good or OK right now.
Moving Forward
If you are really obsessively upset by it, book times to be upset by it and times not to. E.g. On the school run I will pay 100% attention to my DC and enjoy our walk/drive together. At home'on the way to work, I will set the timer on my phone for 10 mins to completely indulge in my self-hate fest about how awful it was to have fallen out with someone. When the timer goes off I'll get up and do something which takes my mind off it.
Pay the Debt
Do something that symbolically helps you feel like you have repaid the 'debt' of having fallen out with someone. Could be doing something for that friend, or for someone else, or in the community. Make sure that it's a finite thing - a single action like babysitting for a night, or helping at a charity jumble sale, or similar, and when it's finished you can say: right, paid off now, so you can move on.
HTH