I agree with all the other supportive comments here, that yes, this was a big fuck up on your part but, you've done everything humanly possible to make sure it doesn't happen again, and to make restitution. You didn't wait for them to come to you, (you went to them and admitted your crime, that has to count for something), You're on medication, you're under medical care, you've done everything that they've asked you to do.
You show true contrition, and you seem honestly like you're trying to make things right.
I certainly hope that whomever it is who you're speaking to who has any kind of final say on what kind of sentence you may receive, can see how much you're trying to do to make up for your laps of judgment which again, gambling is an addiction, and most people who commit a crime while under the influence of their addictive element are not usually punished as seriously, if that at all helps.
I know you must be worried out of your mind, but at this point, I would say take care of yourself, try your best to do so, take care of your children, if there's any more proof that you can gather up to show whatever they need, either what you took, where it was spent, or if you have any journal or diary entries that you might have written at the time to yourself? Something that shows that you were in a really bad place, something that shows your mental state at the time? Anything that you wrote in a journal/diary afterwards saying about how sorry you were and how you want to do everything you can to make it right? Cuz that's got to count for something too right?
Don't know if you have such documents but they probably would be helpful if you did....
You can't change the past, focus on gathering any evidence you can find that they may need so at least you're prepared, and yeah, you will want to tell your kids an age appropriate version of what's going on, you don't want to add more stress into their lives especially pre-Christmas but you know what it doesn't really matter. Stuff happens to people all throughout the year.... At the ages you said your kids are, I think they're old enough to get it.
I'd be very surprised if at the age of 12 and 15 your sons had never nicked anything out of a store even once, almost all kids do by that age, so they probably get it, too. Hopefully they don't understand the addictive element, but they can understand that what you did was a problem, that you tried to get help, that you're trying to make amends, but there will be some kind of punishment involved. They need to understand this, it's only fair for them.