Will a judge honestly make my kids stay when he never sees/speaks to them while I am struggling to make a life with my kids elsewhere?

The children’s father has only seen kids one weekend in nearly two years. There is no financial support. I, the mother, and his brother have begged him to be a part of their lives. He recently filed for divorce wanting 50/50, no child support and has claimed I have alienated him from kids. Our 14 year old knows this isn’t true. I am currently living at my ex mother in law’s home. My two boys and I share one bedroom. I don’t make enough to qualify for low income housing. My significant other just moved to another state for a significant pay raise. He wants us to move in with him. I cannot make it on my own as a single mother and I want to move with the person I love and is willing to step up with raising my kids and supporting us. It is my only way to start a real life for my boys and I. Legal aid says that I don’t qualify because proceeding has started. I make less than $500/month. I cannot afford an attorney at all. With this move, my kids will finally have a real life. What can I do to make this move approved by a judge? My kids and I need this desperately but my ex will fight me on it just out of spite. My son called him, left a message begging for him to call him back. Few days later, he changed his number. I am the parent that drives my kids to/from school. I bring my oldest to counseling once a week, probation check ins twice a month and I do house work with them. He is an absent father with enough power to stop me and my kids living a life they deserve just because he wants to stick it to me. My 14 year old is ok with the move. He even told his counselor he is good with moving and he doesn’t want his dad to have any sort of custody. Which I assured him his father would turn around and loves him. He’s just making poor choices. I am willing to allow Christmases, longer summers etc. with their father assuming he truly wants to be with them. I am not trying to keep him from his kids.