B.,
Consistency is key with kids. Whatever visitation you were granted by the court, you should follow. If you live too far away to take full advantage, start making plans right away to move. Your boys need to know that their mom was there every single chance the court let her. As for your daughter, kids learn how to be a responsible adult not only from the way they were treated by their parent but also the way their parent treated their siblings. Yes, moving is hard, but she will survive it just fine, especially when she is able to see her brothers more often and she sees her mom being a good mom to her brothers.
I have two step-children. When they lived with their mom, my husband would drive 1.5 hours one way to see then for the 2 hours he was allowed in the custody agreement on Wednesday evenings. He never missed a week. He also had custody of them every other weekend. Again, never missed a week. This went on for several years. Later, he got custody of the kids. The kids' mother never consistently saw the kids and the effect on them was devastating. They began to distrust her and as you may know, kids think about themselves what they think about their parents. Eventually, their mom stopped seeing them altogether for 5 years. They both have terrible abandonment issues. One is 18 now and one is 16.
Your kids are still very young. You have a little time to arrange your life so that you can be a part of their life. They need you no matter what problems you have. They don't care if you're broke or depressed. They just want to see you and feel your love. All kids crave this. It's going to be a hard life for the next years til they are grown but you CAN do this.
When my step-kids' mother stopped seeing them regularly and then altogether, I could not help but think that it was a blessing because she brought so much conflict to all of our lives. But it was far, far worse when she quit them. My 18 year old step-daughter still cries over the abandonment and my 16 year old step-son still refers to how his mom treats him like trash. I have extensive examples of what this has done to them but I know all kids are different and I don't want to totally freak you out. Your boys may never have the extreme reaction that my step-kids do but B., do not take that chance.
My husband and I have tried any number of things to help the kids with the problems they have that stem from their "mom issues". They love us and we them. But there is nothing that can replace a mother's love.
I have tried and tried to think of a way to soften this message up as much as possible because I know you are not in an easy situation. And that it may seem impossible to do what you have to to stay in the boys' lives. And I really feel for you and will be praying for you. But you really have to do this. For yourself, the boys and your daughter. Whatever you choose will definitely have a permanent effect on all of you.
Take care of yourself and get some support from your family members and friends. It's going to be hard but again, you CAN DO THIS!
Hope this helps,
S.