Urgh, I remember those days well. Everything is just in shades of grey and you just seem to ache constantly. So much sympathy for you right now.
I'd perhaps be selective about how you read the advice to sleep train. It works for some kids, not for others. I've got two, and it worked for one, not the other. In my experience, if it doesn't work for your kid, in addition to being sleep deprived, you end up feeling more crap. And the Sarah Ockwell Smith book made me feel guilty for wanting to sleep train. (I find her books to be sanctimonious and devoid of actual practical advice.)
I do agree with everyone that says your first priority is to get as much sleep as possible, however that happens. Your husband may work long hours, but he definitely has the easier ride here, so get him to do all housework. Both weekend mornings are yours to catch up on sleep. They are not lie ins, they are replacing the sleep you've not had. Lots of screen time so you can doze.
Gro clock or bribery for your eldest? Although I remember being sleep deprived and bellowing "BED" multiple times when my eldest was that age because I was so broken (although to be fair, it often worked!)
And most importantly, you are doing well. You're keeping your children safe, fed, clean and loved while being absolutely exhausted. You are doing bloody well.