My response assumes that you're about 20 (based on the 89?). I also had 2 kids when I was very young (twins at 22) and I was living with my boyfriend (their dad). So, what I say is not meant to be judgmental or condescending at all, just trying to share what I've learned in the 14 years since then.
It is scary to be young, on your own in the world with 2 kids depending on you, and wondering - based on your last relationship - whether you can trust yourself to make good decisions about men.
When you find a boyfriend who really likes your kids and is good with them, it feels like such a relief - like this is how things are supposed to be. It is the most tempting thing in the world to move in together before he commits to anything, assuming that commitment will come in time. When it doesn't, it feels excruciating to consider ending the relationship, being alone again, and having one more "failed" relationship to make you doubt your own choices.
It is much easier to try to reason with the guy; try to convince him that he should want to make a commitment; try to convince yourself that he really does want to, he's just inhibited by some irrational fear; to seek support from others who will tell you yes, he ought to commit. If you can convince yourself you're "right" about that, you can justify spending more time in the relative security of your current living arrangement, waiting for the light bulb to go on over his head.
It is E.A.S.Y. for a young guy to "play house" with a girl he likes who has cute little kids. Easy. It is not the same as him feeling like a father or being committed. A man who wants to be committed to you and wants to assume a fatherly role toward your kids will ask you to marry him. You won't have to convince him he should ask.
This 23-year-old guy obviously likes you and your kids. So the path of least resistance for him is to tell you what you want to hear. But instead, he's being honest with you that he feels too young to commit and he doesn't think of your kids as his, he isn't ready to be a parent. He means it. That doesn't for one second mean you are not lovable, or that you'll always be on your own with the kids. But it may be several more years before the average guy your age is ready to be a parent.
And certainly it's not fair to your kids that they've gotten attached to this guy and he won't commit to them. But that is your doing, not his. You and only you are their parent (since you say bio-dad is largely uninvolved). Any time you move them in with someone before there's a commitment, you create the risk that they'll get attached and then be hurt and confused and feel disposable. The guy is not responsible for making good decisions for them. Only you are.
I found it invaluable to spend several years not in a relationship - and not looking for one - just focusing on raising my kids and making a life for myself (work, friends, interests, school) that made me feel secure and happy enough that I could say, "If the right guy comes along, great. If I'm alone until the kids are grown up - or forever - I still like the life I'm living." When the right guy did come along, I didn't have to wonder, "Am I attracted to him for the right reasons, or just because I need a partner?" I wasn't in any rush to live with him - my life and my home were just fine, as they were. My kids got to know him as just a friend I was hanging out with, not as an instant step-parent. Once we all got to know each other, he was clear, certain and assertive about wanting to get married and be a family, permanently.
It's really hard at first to do that, to wait for that. But I hope you can find the strength to accept your current boyfriend's position about his own life, make an independent life for yourself and your kids, and then be open to someone new, who knows he wants to be Daddy to your kids, who doesn't have to be talked into it. You all deserve that.