Congratulations on everything you've accomplished! It appears that you've made the choice to be your own person rather than owned by addiction and that's a very hard road to have traveled. You have gone through what had to challenge you on levels most people could never comprehend and I respect you for that. I admire your strength
There's no knowing what your children have been told and you can't really argue the case anyway. People have a way of seeing things just from their point of view, but what matters most is the truth of 'seeing with your own eyes' - in this case, your kids need to learn that you are well, you are clean, sober and on the right path and the only way that matters is with their own eyes.
When you have the opportunity to talk to your eldest, ask him to help you. He can become your greatest support in the family. Ask him what he believes the other children think. Ask him how you might reach each one as an individual.
First and foremost, start with an apology and steer clear of excuses. IE: "Honey, I'm very sorry for not being the mom you deserve. I want another chance and even though you do not owe me that, I believe your heart is big enough to let me try. Will you do that?"
Each child deserves to hear that and with their elder sibling encouraging them to listen you should at least have that door open to you.
Be patient above all. Isn't it better to wait a year and achieve a goal than to rush it and have to wait even longer?
Be honest when they ask about your addiction and incarceration. Keep it simple though. Details can come later. Keep the focus on them and what you hope to have with them for a lifetime to come now that you are the sole owner of yourself
Family counseling can be a wonderful tool, not just for your own interaction with them, but even for their interactions with each other.
I know you want to just erase the mistakes and forget them, moving forward, but amends must be made and like most of us, you have to prove yourself. Unlike most of us, you have to work harder at it.
And whether you know this already or it's something you'll eventually come to know - what you've done to reclaim yourself and what you're doing to rebuild yourself will make you stronger than most of us. I said it earlier and I'll repeat it often, I admire you.