I live with someone who has an anxiety disorder, and in the beginning, when it was not well regulated by medication and therapy things could be tricky, just as you describe.
Your partner's behaviour is abusive if he flies off the handle and shouts at you. Anxiety or not, the behaviour is what it is. It is so easy to go down the rabbit hole of seeking some kind of justification or excuse for unkind, inconsiderate, aggressive, moody behaviour, as in "...but he's struggling with anxiety". Both are true; yes, the behaviour is unacceptable, and yes, he is most likely really suffering. An anxiety disorder is crippling, and unrelenting, always grinding away under the surface, even during times of apparent relief. As a partner of someone whose anxiety causes them to behave like this, the resentment, hurt and exhaustion from the constant worry about what it's going to throw at you next, the constant treading on eggshells, living life on a knife's edge is equally crippling. It sets you up with impossible choices, as you have illustrated so well with your experience of trying to juggle work and the supervision of your children. You play fair, sacrifice, accomodate, seek solutions; anxiety moves the goalposts and blames you in the process, throwing it all back in your face.
I found that keeping DP's anxiety a 'secret' when it first surfaced did me no favours whatsoever. One of the most important things I did for myself early on was to decide that I could tell whoever I wanted, as it impacted my life. So everyone knows, including my employer, our friends, families, GP. Should there be a blip, I won't have to explain, or lie.
It is also important not to buy into a hierarchy of suffering in your house; his diagnosis does not give him a free pass. Your lived experience of feeling worried and beleaguered is of equal importance. Your children, who are sure to feel uncomfortable, insecure and worried, trump everything. For their sake (and yours), he needs to put some boundaries around how he manages his anxiety at home, recognising triggers in time and taking action to remove himself from the situation without acting out, creating a scene or making you caretaker of his needs. I found that my capacity for empathy dramatically improved once DP learnt to take responsibility for himself and didn't involve me in, or make me in any way responsible for his anxiety.
Living the way you describe is awful, and unsustainable. You absolutely do not have to live like this. Don't think for a minute that you can't leave because he has this diagnosis. Your DH needs to see his GP again to review his meds and perhaps ask to be referred for therapy. Think about boundary setting; your DH needs to 'get it' for himself, that he needs to shield you and the DCs from aggressive expressions of his anxiety, and you need to know where your bottom line is.
For me, it really is a day at a time. I know that, although it is well managed at the moment, and DP is very invested in maintaining wellness, things can change. If they do, I will reassess. We both know this. The price of harmony in our house is less spontaneity, more forward planning and careful communication.