You and your husband aren't on the same page at all. That's a red flag. Blended families are tricky anyway and they take time, but it sounds like you have totally different parenting standards.
It's too bad this wasn't addressed before the wedding, but the horse is out of the barn on that one so all you can do is go forward. I'd start with marriage counseling and maybe family counseling.
Your 12 year old isn't "disrespecting the home" - she's being a normal tween! Your husband will find this out when his 6 year old is 12. Does the youngest live with you primarily? Or is she just coming in on weekends and so she's not a daily parenting chore for him?
Your husband can either lighten up and be glad the dishes are in the sink, or he can put a little sign on the faucet that says "Dishwasher?" or he can parent his own child and leave the discipline and rules of the older girls to you. He feels he shouldn't have to tell her - and that's fine - telling her becomes your job.
My concern is that he asks why she is the way she is. Is this a veiled criticism of you and your parenting? I'd be in his face about that!
This is not about dishwashers and plates. It isn't. It's about bigger things - who's the boss, what's normal for a tween/teen, what "power" he may feel has has lost by joining your home, or some issue he has from his past. I don't know which, but you need to get to the bottom of it.
He thinks nagging and tension are a good dynamic for this family? That's not going to work. Sounds like you know it. But if your solution of "Go tell her yourself" is to put it on him, he's going to use his own style with her, which doesn't sound too nurturing or patient. I think that creating a hostile environment with a teenager is a very dangerous path - she'll go find some other guy who is nice to her, or she will find someone who treats her just as aggressively as your husband does and she'll wind up being abused.
If you have anxiety that needs to be treated - go get treatment. No shame in that. But if you don't solve the underlying problem of vastly different parenting styles, this isn't going to go away. You either need to agree with each other on how you are going to handle discipline (and which "offenses" are discipline-worthy and which should be ignored), or he needs to keep his mouth of your child and you keep your mouth off his.
Good luck - but please don't delay.