My son has a counselor. He has serious health issues, as well as everyday living stuff, but, he is able to talk about his fears. We address directly and reinforce his independent thinking about his father and I.
When his dad acts weird I make sure to tell him it is not him. He has a strong base of understanding that his fathers moods, or disconnection are his fathers responsibility, and that my frustration is mine, and that I am to handle that. He does not need to fix his dad, nor is he responsible for "buoying" my mood, or quelling my sadness.
He is pretty darned clear about stuff, but I have to say it is a lot of vigilance and work.
The therapist is very helpful. He has learned to express his worry, and it does seem that just saying a fear or concern out loud, being heard in a safe environment helps him to let it go, and he has a pretty carefree disposition.
You can give that to your kids.
My personal experience is that acting like everything is just fine is what creates the ACoA complex. It is the secrecy, the constant sense in a child, of KNOWING that something is not right, but no one addresses that is part of what causes the anxiety and clinginess... just a general sense of unsureness and distrust of the world, and the childs perceptions.
When we speak openly, within reason, and age appropriately, to the kids, they feel addressed. This alone can create a sense of clarity and a sense that they are not expected to keep any secrets or keep anything inside, fix things, or whatever.
Now my RASTBexH is sober, and he is still unsteady. We address this, as well. His mood swings are his, his to navigate and if he fails to connect or flat out lets son down, I make sure to bolster that that is his fathers choice, and his father loses. It is not about "You".
It took a while to get the language right, and now, at 7, our son knows what alcoholism is. Its not some heavy big word, or tense thing. We talk about it like diabetes, or if his dad has an anxiety issue, I address that, as well. He is really thriving, now.
Just some ideas. These have worked for us. Maybe consult a counselor, you can get free help for your kids by looking into community services.