I know you don't feel like you want to be with anyone else, and I think you're very wise not to pin your future happiness to that, or to go looking for a new partner when you're still dealing with the fall out from the end of your marriage. I promise you, it's not set in stone that you will feel this way forever. I found the idea of another relationship unimaginable when I split from exP. But with time, you may well feel differently, and find someone else to share the good times with.
But, back to now. Let's look at practical stuff. Is there any way of making the future look less terrifying? It must be bloody tough on your own with three. I have two and I'm on my own with them lots of the time. However there may be different ways of making things work better for you. I find it helps me massively to have a plan, something I'm working towards. What's the thing you need to change most? (Maybe getting a part time job or moving?)
What was your job before and did you enjoy it? Do you think it will be possible to get back into it when your youngest is a bit older, or would you need another job? Are you at home now because of choice, or would you rather be working?
Please don't be ashamed, you have done nothing to be ashamed of! And there is no shame in needing to claim benefits, they are a safety net, you and your ex have paid in, now you and the DCs need them, that's what they're there for.