Oh OP, you could be me. Sounds like we work in a similar environment, with similar clients. I am also similarly overworked and despite being senior, am operationally all over an account doing work that a junior should be doing because I care about getting it right.
Firstly, I had some amazing advice a while ago. Fast forward 6 months, how will you feel about this? This is a tiny blip in an otherwise amazing period of producing incredible work for clients that aren't paying the going rate. They will know how much you care, and they will be delighted to have you on their account. This will not linger, even though it feels catastrophic right now.
My advice for right now is get in front of it with the client who doesn't know. Tell them what happened, explain how you acted with the best of intentions and you're incredibly sorry, but an error was made and then list how you're rectifying it. Also point out how you'll ensure it can't happen again. Be honest, be accountable and then, as other have said, let it move on.
Sounds like Client B is acting like a tit and overreacting, but it could that right now, she's terrified about the future for her charity and her job and her mortgage, and you just got both barrels because of it. She'll calm down and know that in the grand scheme of things, it's tiny.
I know you'll feel like shit tonight - I'm just the same and will dwell on all the things that went wrong. Try, if you can, instead of beating yourself up, but listing all the things that went well and all the times you've made a very real difference - it will make you feel so much better.
And then eat chocolate and drink wine.