He's a shit parent, & sees you as his childrens' nanny.
You are far from a wicked stepmother.
You are the 'new partner' we all read about here when the OP is the ex-wife. You know - the one her ex-H cannot do without, because it would not occur to him to parent his own children, let alone develop a fulfilling relationship with them.
DP never takes DSC out without me, ever, nor is he keen for me to go out and do my own thing when they come.
Why, in the name of holy fuck, are you asking his permission to go out solo?
I've suggested he switches it up a bit and actually spends some quality time with them, days out, fun activities etc. Father and son/daughter time. He takes this as a rejection on my part.
Aaaaaw.
Your manbaby isn't keen for you to leave the house under your own steam, feels all rejected if you dare to gently challenge the status quo, & is happy to withhold fun, hobbies, & days out from his kids.
In fact, anything other than "sit quietly staring at your screens & don't bother me until it's time for you to go back to your mother's" seems to give him conniptions.
Does he even cook for them - or is that down to you?
Ever engage with e.g. parents evening, school events - or is that all down to their mum?
Is he usually this bone idle, selfish & ignorant, or do these traits only emerge when his kids show up (go on, surprise me)?
As talking to him doesn't appear to have worked, you are gonna have to show him.
On a contact evening, just go out.
He will react with horror, bafflement, & manipulations.
Do not engage. All you need is a breezy "popping round to Sue's, be back around 9:30" or whatever.
On a contact weekend, ask the kids if they'd like to go out for a burger, or swimming, (whatever) - & when he starts fussing, tell him he's welcome to join you, but you are bored rigid, so are the kids, & everybody needs to have funtime away from their screens.
Do it OP.
If nothing else it'll be a valuable experiment for you.