How do you handle finances as a part time worker and your spouse makes significantly more than you and holds it over your head?

You need to sit down and have a long serious conversation. Dave Ramsey actually has good way of explaining where both your salaries are each other's. It's a "WE" make xxx.
Tell him how it makes your feel and ask him if he has any suggestions on how to fix this. This will make him feel more secure and less attacked. Then offer some suggestions or thoughts, join account with 50% of each of your salaries going into. Or a budget of 200 or some other amount that can be spent without informing the other

I just want to add, I saw this with my parents and 25 years later nothing has changed so for your sanity try and work this through now

Tell him you are going to get a job and see how he reacts. Sounds like he's very happy your trapped and does not want you to have any independence. He won't give you money? Make you own. Of course he will kick off or find a way to sabotage your independence but this is where you see if he's a man who holds you up like a good one and is proud of your successes or tries to hold you down because you are a slave and only he can succeed.

We just share money. We chose to share a life, a house, cat, cars and children and money just naturally came into that. We never had an official chat about it but it’s what we both do. We’re not married but have been together 15 years.

If we weren’t like this and I went part time. I would make sure that he was paying the bills and giving money to cover child costs. It is not fair for one person to give up their salary to look after the children and the other person to not support them financially. I would ask for the money to go into a joint account if my partner was working and I was solely looking after his children. You should not have to ask for spending money.

My husband makes about double I do, we have a joint account for everything that’s for the house or the kids and we split it 60-40 ish. Then what we have left is our own. But when I run out of my own money I ask him for more, because ultimately I’m his wife and we decided together that I’ll be the default parent etc. we are a partnership in everything

That's financial abuse. Definitely talk to him about it.