Time for you to set and enforce reasonable behavioral standards in you home including giving your DH clarity that he will not undermine you as his equity life partner or as an equity parent in your marriage and blended family.

If he fails to immediately and consistently support you then file.

Take care of you

As you know, President Lincoln said: "A house divided against itself cannot stand ..." That quote popped into my head when I read your post.

You and DH need to get on the same page. Is DH aware of the difference in your parenting styles? Have you spoken about your different approaches to life and the consequences of this disconnect? Sounds silly, but I am always amazed at how that giant elephant in the living room can be ignored. You may not be right, or you may be 100% spot-on. Or, you may be insisting upon a parenting style which matches your preconceived ideas of "should." DH may refuse to address things because he is afraid of inflicting more 'harm' to his children. Does DH see his own undercutting of your authority in the family? Too bad that DH is unable to discipline his own children. Fear is a huge factor in parenting and step-parenting.

Marital/family counseling? Parenting class? In person, in paperback book form or online, there are lots of resources to help all of us become better parents and partners.

BTW: you have the tween/early teen set in your house right now. Can you imagine the 'full-on press' once all three kids hit their high school years? {Hugs}

You have no idea what you are in for as these kids become adults, poor lady. It gets worse, as hard as it might be to believe; even nastier. Fortunately, he supports my never being around the rudeness he created (and he knows they are messed up though would never admit it), so I am at peace 99.9 of the time now. If I had to live with this dynamic constantly or (even be around it), I would file for divorce, no other choice.

You have to protect yourself, and you have nobody else who will do it for you, it appears. Many of us have doormat husbands, they rarely change. They never step up, it is sickening for a wife to watch and experience. Your only hope is to concentrate on your own health and happiness. Do whatever it takes to make yourself happy!

Remember your husband is lugging around a load of guilt for three kids, especially if he chose an abuser to be their mom. So you have two problems: the kids and your husband.
I agree counseling may help you deal with your husband and husband deal with his kids - which is a better than you dealing with his kids and also having to deal with his road blocking on the side (you'd be fighting two battles at once!).
You may have different parenting styles, or maybe he's so guilt ridden he just can't bring himself to discipline them the way he must. But I think the key is dealing with husband in counseling first to find out if it's style or history holding him back. If the counseling goes well with husband, it will make parenting the three kids easier.
I know counseling seems like a long slow haul when you just want one peaceful night. We tried to deal with overriding issues in counseling, but also used some time in each session to deal with a specific example of where husband was failing. It goes us some immediate relief when similar situations came up, but worked on the big picture too. If you husband is motivated for improvement, you'll save on the divorce lawyer. Good luck - we all know how hard it is.

Three teenagers, OMG!!! :jawdrop:
Hard work, patience, and you are not the Mother! so, that´s even more difficult.
It will take time to adjust, even to start a routine. So before filing, consider getting extra help. Have you considered an "au pair"(nanny coming from abroad?) https://www.aupairworld.com
She can help with the house and you organize the rest.
A deep talk with DH to set up bounderies, counseling and support from the family also would help. These kids lived a nightmare in the past, and now they might be lost and not understanding what is really going on (also they might be angry, resentful... and it´s not your fault)
I know it´s too much for you, honey. Take a deep breath, devote some time for yourself and practice any sport or technique to help you to gain autocontrol : Yoga, Reiki, breathing exercise.
So, you´ve got a big gift this year... Believe me anyone will be shocked with 3 teenagers at home.