Christ, some of the responses on this thread 

There's nothing wrong with wanting a relationship ffs. And admitting feeling lonely despite a full life is really brave and honest. I actually am more sceptical of the people who claim to have been single for 10+ years and are perfectly happy, never feel alone etc.
I'm not single now but I was for about 6 years. And that was hard. Yes sometimes it was lovely being on my own. I'd been in an abusive relationship and quite miserable for several years. Having the freedom to do my own thing, live how I wanted with me and my DC was lovely.
But as time passed, I didn't just want that. I wanted a good relationship, a true partnership of equals. So I dated a lot. I put myself out there a lot. I expect the OP has too. But nice though it is to think that works, it makes no real difference. It's luck and chance more than anything.
It matters not what you look like, what size you are, your financial position, educational level, etc etc. We tell ourselves it does. Or give that advice to others, because we want it to be true. It really isn't. Ultimately you just have to be in the right place at the right time.
There is no magic formula for meeting someone. I met my bf completely by chance. His profile popped up on an OD site. He wasn't really what I was looking for but he seemed nice enough so I sent him a message. But that wasn't doing anything different, I had sent the first message before, just rarely got a response. I got stood up the week before I met bf (who knows had that guy turned up we might not have met).
And when we met, I was looking for a relationship. That was what I wanted. I wanted something that you can't get from DC or friends. I wanted someone to care about me, and who I could care about. A companion. Someone who would be on my side, listen to me. Be interested in me. Oh and who would (to quote someone I once worked with) give me a good seeing to.
Yes you can do ONS. Apparently you can also do FWB but I always feel they're an urban myth. Everyone says they have them but do they really exist...? But ONS in my experience are no solution. They might in the very short term relieve a sexual urge, but they dont help with any desire for intimacy, or the physical but not sexual side of a relationship iyswim. My bf often rubs my back to help me sleep...how else would I get that outside a relationship?
That's not the only reason we're together of course. But it's something I couldn't have had when single. However full my life was, and it was. When you're a single parent your life is always full. But that doesn't stop you feeling lonely. Whether it's at night when you've put young children to bed. Or if you have older DC in the early evening when they're glued to phones/gaming, or out with friends. Friends only can do so much, ditto family.
I'm sorry there's no answer to this. All I can offer OP and others in the same boat is that your feelings are perfectly valid and natural. It doesn't mean you have low self esteem or don't love yourself. Or that you are wrong for not filling your life more, or doing more, or putting yourself out there (all of which I expect you do or have done). I can't offer any words of wisdom, only that I've been there and I get it 