TBH I think you are completely overestimating the amount of concern that would come if you ask for help. Ideally if someone was not managing there WOULD be some kind of intensive intervention in order to help you get back onto a more even footing and sort of "reset" things, because it is hard to cycle back from the state where the children are chaotic and attention-seeking, and you are reactive because you don't have the mental energy to be proactive and because it's the only thing which sometimes works. However, in reality there just are not the resources to support every family so resources go to families who are much further into crisis than you are.
But there is a huge amount of in between steps from no support at all because nobody knows you're struggling, to people coming into your home and telling you what to do or (god forbid) removal of children. None of these are especially intrusive but they also aren't likely to become available unless you can communicate to someone that you're finding it hard.
I'm talking about things like antidepressants or anti-anxiety medication as someone else said, funding for extra hours at childcare to give you more of a break or allow for more 1:1 time with each child separately, parenting classes (I know you don't want parenting advice, but it's something which may be offered and sometimes having a chance to discuss with someone IRL can help, you can also refuse), CBT or other therapy for you, a family support worker to help with tasks like housekeeping or just be an extra pair of hands.
You are saying I am at breaking point...they are unmanageable together...I tense up and things [are] more likely to go wrong...I’ve become the most horrible mother lately. I shout; I say things that are awful ... I’ve cried, they’ve cried. It’s horrible.
Something has got to change. I completely hear you saying that you don't have the energy or bandwidth to be driving change. This is the point where it makes sense to say you know what, I've reached my limit, it's not getting better on its own - I need to ask for help. Try something different, even if it is scary, because what you have tried so far has not worked, that is not your fault, you don't need to accept it as some kind of punishment.
As you've said, most threads with people struggling are with the youngest child under one, which means your situation is different to the norm. Your DH's injury is also exacerbating things. Two children is genuinely stressful, many people struggle. But most people find their way into a routine, it's really OK to ask for help when that hasn't happened. We can't do everything on our own. It is not a personal failure of yours to find things hard - we all find different things hard, and not everyone has the same set of circumstances. Sometimes we don't even know how our own circumstances compare to somebody else's.
My eldest DS was extremely challenging but I had no idea. I just thought it was normal parenting and I was shit at it. I really really wish in hindsight that I had reached out for support. When DS2 was born I realised how different it was to the experience with DS1, and then later on when DS2 started to struggle in different ways, I did look for support, and it has been so different. Even though the actual "support" (ie, intervention) has been laughably minimal.
It looks to me like you have two blocks: You think using fear-based discipline might improve behaviour, but you don't want to do this because it is no longer advised. I think this instinct is good; fear-based discipline can result in more compliance in the moment, but tends to increase aggression overall and is not effective to improve behaviour long term. It is likely all the things that parents have always done outside the fear-based discipline which changes behaviour long term. I don't think this would help your situation, because you don't have the other, balancing structures in place. However, you are already veering into this sometimes by shouting, or by "trying to shock them". It's understandable, but it's also the thing you say you don't want to do.
The other block is asking for help, you don't want to do this because you are afraid of, I think judgement? That it feels like admitting you have failed or are weak? Or something happening which you might not be able to control. The thing is, you are already not in control of the situation, you keep saying how miserable it is and how impossible you find it and how you know that your reactions are counterproductive but you are not able to react differently in the moment (which I completely understand, as I have been there). And I also think that the level of intervention you fear is unlikely, or very far away from where you currently are. I wish I could guarantee a sympathetic reception for you for this, because I think that would make a lot of difference. Unfortunately I don't know how easy that is to find, so I do understand how difficult it is to reach out, but I think this is about putting on your own oxygen mask first.
Something has to give, so which one are you going to choose? Are you going to opt for using fear, or asking for help?