C.L.
Sounds like your mom is simply trying to protect you and your children. Although, I highly doubt your mother would be able to get custody of your children, unless you are into drugs or something else that is not being said.
My suggestion, is not to move in with the new boyfriend. You need to grow up and live on your own for a while. Get your own apartment/house and date this guy, or any guy for that matter for at least a year before you even talk about living with them. Keep in mind, that you do have these kids to take care of too, and shuffling them from one boyfriends house to another when it doesn't work out isn't the way to go.
I know, I'm probably not saying what you want to hear, but I've seen too many people make the same repeated mistakes and their children suffer for it.