I'm going to tell a little story. it's a different view point and may help to understand how this might work for your kids.
In the viewpoint of the kids, of course you know what's best for them as their mom and a wonderful mom you are. I am going a child's perspective from the live-in MLC situation.
My mother was not in MLC, she was in a mental health crisis. her behaviors very much mimicked your H's. I lived with this my whole life until senior year of high school when my dad finally left. Only I was left alone with her.
My dad stayed for me all those years. he thought maybe he could cover up the effect my mother had one me by being the great parent. he was a great parent and is probably the reason why I am not on the street doing drugs or something bad.
But looking back, I was in a lot of pain watching my mom. My dad made my life full, but he didn't cover up the absence of my mother. I was hurting when she chose not to cjoin the family in almost anything. I was hurt watching her sleep all the time. I was hurt when to chose to go to Bingo every night (she traded a drug addiction for a gambling addiction)When she had no desire to eat as a family or would go and hide somewhere. The toll it was taking on my dad to have an absent wife was very very evident.
It was painful for me to watch. it also left some "I'm not worthy of my mother's love and attention" damage.
If you decide to take you and your boys out of this situation, it may actually be the better option for them. They see what's going on. They feel what's going on. Like I said, my father was a wonderful engaged father as you are a wonderful engaged mother. But this is right in front of their faces no matter how much you try to compensate.
It's something to just think about. If anything, I don't think you would be harming their childhoods if you decided to get out of this situation. There may actually be an improvement or a relief.
I hate to say it, but I kind of wish my dad had left and taken me with him. I would visit my mom of course, but the situation was so obvious especially as I got older that it became too uncomfortable for me and I took it personally. I spent majority of my time out of the house and at friends just to avoid it. Or in my room just reading books. Or sleeping. I slept a lot to avoid it.
I'm so sorry this is happening to you all. it's much akin to living with someone with a mental illness and that is difficult thing to do.