You got 3 years out of a relationship. That's solid. If she really wants kids and you don't want them, you're going to have to end it now before it ends badly. You're going to have to have a matter-of-fact conversation and end it.
You're in a tough spot at age 34. If you're dating women 29-34, a lot of them are starting to feel rumbling ovaries if they are childless. It's not easy to meet a woman 28-39 who is childless and doesn't want children. When a woman turns 40 and is childless, she might been more inclined to accept the reality that she isn't going to have children and is willing to live a childless dating life with a man.
I have never been all too enthusiastic about having children at any point in my dating history. At 39, it'd be tough for me to start a life with children. It would impact my retirement if I were to have children in my 40s. Additionally, unless I was having children with a woman under 30, there are higher risks of birth defects or other abnormalities. Two older parents isn't best for a child's developmental health.
If you're a 30 something man looking to date 30 something women and you don't want kids, you're in a tough spot. If you say right away you don't want kids, your prospect pool will diminish to essentially zero. I would recommend saying you're open to children in the future, but it's not a short term priority. That way, you'll keep prospects around. Try to date 24-29 year olds if you can for the next few years. The only 30 somethings worth dating are the ones who are childless and don't want kids in your position.