"I agreed to have them live with us because I wanted to MAKE HIM HAPPY that his kids were not farmed off to strangers."
Thereby lies the problem. You wanted to make DH happy, which makes you happy. Perhaps you wanted to play the heroine role; I don't know.
You don't appear to have done it with, "Poor kids, they've been through hell, they need stability and love" in mind, that's coming across loud and clear.
Now they're here and one is a grown man you seek to control them. So what if a 19 yo is on the Xbox for 3 days? How does that affect YOU? it's not a crime! Back off, you cannot control EVERY part of this family and you'll tie yourself in knots trying to.
I know you're going to get vexed with this post. I've read how you've told others to fuck off when they disagreed with you. Some of your posts are incredibly aggressive and defensive. I don't care, it's not my circus, you're not my monkeys, it doesn't affect me. I'm just telling you how you come across to see if that gives some clarity.
You have choices. A). Issue an ultimatum. It's your kids or me. Then be prepared to either be left on your own or to spend a lifetime with the kind of man who abandons his kids. If the latter happens, FFS don't have kids of your own with him.
B). Divorce him, sell the house, work out with a solicitor who gets what and move on. Rented properties, if you need to opt for that although it sounds like you won't, SOMETIMES will accept dogs so don't use that as an excuse.
C. Continue in this ever ridiculous circle of living with a man who is willing to remain in the same house as a woman who hates his kids. Continue to live with those kids and prepare for backlashes when you try to control and micromanage what perfectly legal, albeit a little dull way they have of spending sunny days. Be miserable. Go on, fill your boots. It doesn't affect us.
D. Opt for counselling, family meetings and discussions, be willing to admit that you're not always right, stop controlling, quit the insecurity and give it one last shot. If that fails, refer to B.
One thing it's NOT on to do is to keep going through the same revolving door without ever stepping out and then being obnoxious to people who point that out and offer you a few suggestions or some hometruths,
I hope that one way or another you'll resolve this and that everyone in the family is happy, valued and respected.