How do people manage with multiple children?

Sorry you feel like this, OP.
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Your shouty husband doesn't help you or your kids and (I get bashed for these) if financially possible, definitely better to parent yourself without him
at your home, if kids already present.
Another - life is just so much better and authentic without children (and at my age, probably without men) - applied to myself.

Sorry this sounds so stressful. I second the recommendation for the book mentioned above - I only got through half of it but it’s already been so helpful, lots is about working out why we get so angry at certain situations/behaviours and how to avoid getting into that vortex. Your H might be able to get it as an audio book via the library, I did.

I wonder if he needs to get out more too - go for a run maybe to burn off some of his own frustration.

Getting out of the house helps so much but I know how hard that is with a small baby. I recommend library storytimes or things like aquarium if there is one near you, most babies find it chill and your toddler can run around.

I am so sorry you are going through this. I had a 5 year old 2 year old twins and a newborn. I do think your DH is not helping at all with shouting. It is unacceptable to have him shouting at your son. Shouting can cause children to become aggressive and change behaviour. Look this up and show it to DH. He has to realise the damage he is doing. I think this is affecting you more than your sons behaviour.
As people have said you need to pick your battles. If not going out don't worry about the dressing. If you need to get something done do it like a game timing him to do things. As the weather is bad find things to do indoors. He is the age now that he can do puzzles. Pairs with his favourite characters. Flash cards with colours numbers, letters. Drawing painting things he can do while you are also sitting with baby. Nothing wrong with some TV watching a film together.

Any attention is attention. Make sure you are both giving positive attention for the good things and not just focusing on the bad behaviours. It can be tricky when you have a new baby as they see more attention going to the babies needs ie baby gets cuddle time because she is breastfeeding.

Pretty much every 2/3 yo I know prefers socks off and it is actually quite important for them to have barefoot time as it helps the foot to grow in proper alignment and for their muscles and balance to develop well.

Meals - they join you at the table or they don’t eat. Eat as a family so it is a sociable time. Don’t give snacks as replacements for uneaten meals. Don’t make it a battle, put unfamiliar foods on the plate with foods they eat so they can get used to them.

With young children expectation is key. You expect them to eat at the table, you expect them to put their shoes on to go out. It isn’t a negotiation or a request or a battle. If they don’t put their shoes on go without them (assuming the other adult is home of course). Make it clear you and baby are going to the park and if he wants to come he has to get his shoes on. There will be a few tears the first couple times but he will learn that you mean what you say. I think parenting has become too much asking what the child wants rather them just being told. Choices are fine but they have to be real ones you don’t mind either outcome for ie do you want to wear your red shoes or your blue ones.

Don't forget you have another toddler on the way - why do you think your baby will behave any differently? Maybe suggest you both try your way for a week and see how it goes. Then if you must try his and see how that goes.

Ah, you are doing the hard yards and your three year old's little life has had a bomb into it with the arrival of a baby.

The terrible twos are nothing compared to the frightful frees. However, your DH sounds like the biggest baby of the lot. Could he work from the office?

Agreed.

Clearly the issue is your DH and not your children. How open to finding new parenting approaches is he? Are there any podcasts you could point him to? There is a good book by Philippa Perry called the book you wish your parents had read and also another one called No Bad Kids, I think that is Janet Lansbury.

Many would rather cough up some money ( after all a small percentage of earnings is likely less than they pay to support a household) than have to do 50/50 care

I'm really sorry you're going through this OP. It's an awful feeling doing the best for your young children and your partner's there sabotaging what you want for them when he should be working with you towards giving them a better childhood than his.

Ultimately the ball is in his court. It's all very well saying he's sorry but he needs to actually find a way to do things differently. If I were you I'd be deciding to myself how long I'd give it before enough's enough.

I think he needs some clear boundaries - he needs to make the choice to learn better parenting methods and put them into action otherwise he will be responsible for breaking up his family.

Then the natural consequence is not having 50/50 parenting and having to pay shit loads of maintenance

The 3 year might seem like the big kid now there’s a new baby, but he’s still very little himself and waaay too much is being expected of him. Also his whole world has been recently turned upside down due to the baby so of course he’s not behaving impeccably!

Then he doesn't believe you'll do it.

I think he either needs to engage in some learning if he doesn't have the tools, or simply listen to you and pull his finger out, or if you let it slide, your 3yo could get 'easier' in time. Or obviously, split.

I've no idea why people keep coming out with this. If fathers are not good at looking after kids when with the mother the chances are they will walk away and have no interest in spending time with kids if they split up.
You cannot force a parent to spend time with their kids

Does your DH have to wfh?

Welcome to toddler life!

Does your DH acknowledge his poor behaviour and actually want to commit to changing OP? Because that's the most important first step.

If not, I'm afraid I can't see a way through it as a family unit.

He's picking up on your husbands shitty attitude and that's why he's getting upset/stressed out.

None of this excuses the way you two are behaving towards your 3 year old. Why are you letting your husband behave like that to him?!

Perhaps he needs it spelling out to him, that if you split, he will be expected to look after two children regularly on his OWN instead.

Fucking weasel. He should parent up. It's hugely unattractive.