Im afraid to tell my boyfriend I don’t want to co-sign lease with him. What should I do?

If you plan on living there, you should sign the lease. Why should he be stuck with the financial burden if you decide to break the lease? Most LLs won’t allow anyone else to live there if they aren’t on the lease. He could be violating the lease by having you live there and not sign the lease. If you don’t think you’re gonna make it as a couple, you shouldn’t uproot your children’s lives to move in with this guy. I’m guessing he would only want a 1 bedroom apartment for himself, and he’s probably getting a larger place to accommodate you and your kids. It’s not fair to leave him to pay all the rent for a larger apartment for the rest of the lease, and you just leave without any obligation. That’s not fair to him. It’s also a bit concerning that you’re only worried about being stuck in a lease if you break up, but you have no problem uprooting your kids so you can follow this guy to another state. If you aren’t comfortable signing a lease with him, then you shouldn’t take your kids and follow him out of state. I think you’re being unreasonable to assume that the lease should be his sole responsibility. It’s your responsibility too. That’s a risk you take by living with someone and not being married. It sounds like you may not be ready to live with him. I never recommend moving out of state to follow a man that hasn’t made a commitment to you. Don’t leave your whole life behind to follow a man who hasn’t put a ring on it. 

Why should he be on the hook for all the legal & financial burden? So you want to live there obligation-free and be able to cut & run without any consequences? You're not emotionally mature enough to make that commitment.

You’re not ready to be in a relationship if you can’t even discuss these things with him. It is important. No wonder you two are having issues. 

Speak with him NOW. This can’t wait.