D.B.
I absolutely understand your frustration. You have talked until you are blue in the face, and now you are so frustrated that you feel you have no options except yelling and spanking. But they aren't working, are they? It sounds like your kids are at each other a lot, and they either have a fit (yelling) or they try to get on the toilet seat at the same time (pushing & shoving, I imagine) or one pushes the other over.
The problem you're encountering is that they don't know how to regulate their behavior, or use their words instead of being physical (either against each other or individually by throwing fits). You want them to (ideally) do what they are supposed to do (get a fork, for example) or at least deal with their frustrations calmly. You don't want them to get to the end of their rope and erupt in frustration.
So here's the really really hard part. You have to model the behavior you want to see in them. That means you can't lose your temper, and you can't react by spanking. Otherwise, they see that's what big people do, so that's what they do. I don't want to get into a big argument over whether spanking is okay or not, but the fact is, it's not working for you or your kids.
Instead, you have to validate their feelings but show them a better way to do things. This takes time and patience, and I know you don't feel you have much of either. But if your kids keep seeing that you blow up, they'll do the same thing. One thing that's happening is that A is trying to get your attention by being babyish (saying she's going to fall in the toilet or can't get a fork by herself, for example). So you have to get in her face in a calm way and say, "I know you want me to get your fork right now, but I can't do it until I finish (whatever). So you can decide whether to wait for me or to get it yourself. Your choice." When M pushes A aside or starts something, CALMLY separate them and say they need time apart and to go to their rooms until they can play calmly and nicely. Remove the toy they are fighting over so that no one get to play with it. "I'll bring this back when you are calm and ready to get along." Say this matter-of-factly and without anger, as hard as that is.
When they see that their behavior does not get a reaction (and remember that negative attention is sometimes better, in their minds, than no attention at all), then they will realize their bad behavior does not pay off. When they come back together, give them hugs and tell them you love them. Also, try to "catch them being good" and compliment them when they play nicely or share. Don't say, "Isn't this better than an hour ago when you were fighting?" Just notice the good and remark on it.
You are not the worst mother. You are a tired and frustrated mom who picks her kids up and then tries to get everything done. If you can take 10 minutes for a cup of coffee at a coffee shop between your job and when you pick them up, just to re-group, do so. If you can take 10 minutes to just play and interact with them when you get home before you start dinner or your other responsibilities, do so. I know it's hard but if you can discipline yourself and get into a calmer frame of mind, the kids will follow suit. It won't happen the first day so try really hard not to get sucked back into the yelling/spanking cycle. You know that isn't working so don't go back to it.
Good luck!