You are still in the trenches. It will get better. It won’t happen overnight.
That said, there are a lot of things you can do to make things a little bit easier for yourself now.
You have said that the children are well behaved and eat well at school. That suggests that that changing your behaviour might have an impact on theirs. But also take comfort in the fact that kids act out with the people they love and trust.
Nutrition
I understand your concerns. Really, I do! I hate meal planning for my DC. But…if they eat well at school 5 days a week, they are getting good nutrition. So don’t worry about that.
But you can still make improvements at home. List what will eat without argument. Really, list it. Stick it on the fridge. Refer to it when you need inspiration. Always have those foods in. Don’t worry about ‘meals’ per se. Give them random picky bits if that is what it takes to get them to eat something.
If it bothers you that DH gives them sweet treat, stop buying them. It does sound like you do the grocery shopping and not him so that should be easy to do. Do get other snacks that you are happy with whether that is plain crackers, cheese or fruit. Put them all together in one place. Both the kids and DH will know that is where to go for snacks so your DH doesn’t screw up give them other stuff. They can have ice cream etc as a treat when they go out (or rewears for good behaviour). Also, there is nothing wrong with frozen pizza once a week if it makes your life easier.
If they don’t eat, don’t worry. They are eating well at school so they won’t starve. You just need to get enough food in them to prevent them getting Hangry. So either give them what they will eat or give them what you want them to eat and accept you are going to have a shitty few days until they eat what you give them or go back to school.
Attention
It is fucking exhausting. I get it. Here is the shit bit - you have to put the kids first. They need your attention before you can pay attention to yourself. And it sucks because you know you would be a better parent if you could get yourself sorted out first but you just don’t have the time.
Somebody else suggested love bombing the kids. It is a good idea. Give them loads of attention. Overwhelm them with it. Accept that everything else is on the back burner for a week or two. And then start putting in some time for yourself. Give the kids a heads up about this - I’ll play for twenty minutes but then I have to do whatever. And then prepare them for briefly losing your attention - five more minutes and them I have to do whatever. And when you leave - you keep going with this and I’ll be back in 5 minutes.
Your DH
He sounds a bit crap. As a PP mentioned, a lot of women find it easier to be a single parent than a married to a shit husband parent. It means you have one less person to worry about, you have one less person screwing up your plans, and you have a small chance of every other weekend off duty.
You
Your posts have a lot of ‘I can’t because…’ in them. Actually, you can. You just don’t want to because it is going to be difficult.
And that is actually ok. If you don’t want to try everything suggested here, you don’t have to. But there is no easy fix for your situation.
You need to choose to either accept things as they are semi permanently (until the kids are older and you are out of the trenches) or accept a couple of weeks of shittiness and exhaustion and consequences and behaviour management and consistency and routine in order to change the children’s behaviour at home.
Remember - they can behave themselves in a school setting so they can’t be all that bad and they can learn the same behaviour at home.