I haven't read all the replies so I do apologise if I'm repeating anything.
I'm three years into a relationship with my partner who lives with me and my two sons. I had separated from their dad 18 months before I met current partner.
Like you, I wanted a relationship and felt like the children were considered by some as baggage. My view at the time was that I'd really rather not have any more children of my own but was more than happy to be a step mum to someone else's, since that was what I was expecting someone to do for mine.
I was using dating apps, tinder and hinge. I was open about the fact that I had children but made a point of saying their father was involved, saw them regularly etc. I went out of my way to match with single dads as I figured they'd be less put off by my children. I think I even had a jokey line like "single dads encouraged to apply."
To be honest I didn't really struggle to find matches. I guess by being open and honest about having kids in the first place, those for who it was an issue just swiped by. Only once did I match with a guy who hadnt read my profile properly and he (reasonably politely) backed out because it wasn't for him
For context (and I think this was important ) I was 32 nearly 33 and had my age setting looking for someone 30-40.
Eventually my current partner popped up on hinge with its algorithm saying it thought he and I were compatible. I was disappointed to see he had no children but matched anyway. He was 34. He had no children. We met up and the rest is history.
In terms of "taking them on as his own" I guess it depends what you mean and how involved your child's dad is. Now my partner lives with us in a house we own 50-50. He pays half of all household bills and food shopping so I guess he has some financial contribution to my kids in that respect but I pay all childcare, dinner money, swimming lessons, new clothes etc. When we go on holiday I tend to pay for me and the children and he pays for himself. On odd days out he generally offers to pay half for the four of us rather than just his quarter. I do all school runs and take them to football practice etc but that's mainly because it's in the right direction for my work. When I need to work late or whatever he will do it if I need. If I want to go out running or to the gym or to the shops or four dinner with my mates, he is happy to look after them at home if he is free. I try not to take advantage and tell him regularly how grateful I am.
My children are at their dads two nights a week so we have just us time.
I feel very lucky and have asked him before about why he chose to take us all on. His answer is that he was well into his thirties looking for a partner and realistically if you're looking to go out with a woman in her thirties, there's a reasonable chance that she has had children from a previous relationship. He just took it as a likely thing, not baggage or a negative, just a fairly likely thing. He says he has no desire to have his own children but loves that he can be a positive influence on mine. He loves them very much.
They do exist!