well..... the very best thing you can do for them is to have your own life and be happy. That's not theory btw but practise. They will reacte badly to things not only changing, your cosy trio, but also at their age the idea of you being sexually viable is horrifying. You're supposed to be baking cakes and going to bed at 9, your life's mission and purpose subsumed in them.
You may have led them to believe that OP - you've bent over backwards to help them cope with the upheavals of the past few years. You can't get it right eh, but I did the same and tbh I now believe that was a mistake. Easy to say in retrospect but I needed to be a bit more gung-ho about it all, along the lines of shit happens kids, ah well, what's for tea. You and I have given our kids the impression that a terrible thing has happened to them and they will barely be able to cope unless we are at their side smoothing it all over for them. Don't treat them like hospital patients. Shit does happen, to a lot of people, get over it.
Someone said very similar to me when it was all happening - worse, actually - and I was horrified at her hardness, truly flabbergasted that she didn't understand what was happening to my babies. I now think she did understand, perfectly. I'm not suggesting you should be hard - don't actually say 'get over it'! - but act along those lines. Life as normal type of thing. We're moving on chaps, what's for tea.
You've been used to pandering to an addict husband, mopping up his messes, being dragged around by his dramas, and that has probably established some furrows that need to be dug up. You don't have to do it with your kids (even though you may feel you're redundant if you don't? 
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I'm also concerned that they freeze you out 'for days' when you don't do what they want. This is manipulative behaviour and I'm sorry to say my kids show some very clear signs that they learned a lot from their dad. Don't let them get away with that, but don't say anything about it - don't agonise and plead and sigh and shout - but act like nothing's happening - even, that you are so taken up with your own life/new man that you don't notice. Don't feed it. That will stop any manipulative behaviour in its tracks because at root they are afraid of losing you and your nonchalance will play into that. They are not going to lose you - you know that - but don't let them manipulate you, don't pander to it.
Sorry, long post, strong views. I wish someone had said something like this at the time we were going through it. Alright, someone did, but I didn't understand it and she said it brutally so I didn't listen - but I remembered it. Things in my family are a godawful mess however many years down the line and if only I had started on them earlier. My kids think my life's purpose is to see to their every need and are incandescent that I won't and don't - I did such a good, perfect job of supporting them that I led them to believe that that's what I was on this planet for.