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He’s massively unreasonable.
Look you must know surely that it’s not reasonable for him to insist you spend all of your free time with him and his kids - or watching his kids for him.
They are not your children and therefore not your responsibility. Even if you like them, even if you were married, even if you lived with them 50% of the time - there is nothing that dictates you have to spend all your time with him or them.
Yes if you decide to have a relationship with someone who has kids then you do end up in a family dynamic, it’s impossible not to. But you have to be happy with the level of involvement you have, both parties do.
There’s massive red flags here because when you’ve expressed your feelings, he’s minimised them, gaslighted you and low level threatened to break up with you in order to get his own way.
That’s not what a reasonable person does.
His contact time is for HIM to have contact with his kids, if you’re not comfortable with his level of expectations for your involvement and he won’t back down, well that’s the end of the relationship isn’t it.
Hell would freeze over before I let someone manipulate me like this. Who the fuck does he think he is!
Stand up for yourself, it’s your life OP. Don’t be coerced into a lifestyle you don’t want.