I'm going to sidestep the questioning of single parenting because it's sort of triggering to me.
As for when you should start a new relationship? I would say when you feel you have thoroughly washed the consequences of the old one off every part of your being. Which won't ever happen. But the point is -- you will always carry your experiences of your dysfunctional relationship into the future. You want to get to the point where you can use them as reverse indicators: If this is what you would have done in your OLD relationship, do the OPPOSITE...
I jumped into my next relationship too fast.
There, I said it.
As luck would have it, I didn't jump into another relationship with an alcoholic (my new man has a beer maybe twice a year?). As luck would have it, I didn't jump into another relationship with a manipulative control freak. But I could have.
I don't regret it. We have a great relationship. HOWEVER, he's got to be the most patient guy on earth (and I include those Indian gurus sitting on spike mats in that statement) because dangit, I haven't been easy to be with. Mainly because I still react to HIM as if he was AXH sometimes.
For example: AXH would deliberately punish me by ignoring my birthday because he knew it was important to me to celebrate and he knew it cut me to the core if he didn't acknowledge it. So when my birthday came around and my new man was out of town and in a knock-down, drag-out negotiating meeting all day (and I KNEW all of that) and he didn't send me flowers or a cake or promise to take me out to dinner when he got back -- I thought "he really doesn't care about me. He's deliberately hurting me."
He wasn't. He was just flipping busy, and when he returned, it was with the perfect birthday celebration, a few days late. But my first reaction was irrational and completely colored by my dysfunctional marriage to an A.
I'm telling you the story because I would have liked to sort that kind of BS out BEFORE having to expose someone else to it. Now, I don't know if that's possible -- if maybe you'll always have to adjust your reactions in a new relationship -- but it's exhausting and tiring and wearing on that other person. I'm glad we are together, but I wish I had been healthier before we got together.