That was a beautiful post that you wrote. It read like an open letter to your boys and it showed so much love and warmth for them.
25yearsmlc has it right about your boys and a new woman in your life, but I know, from reading everything here, that you already know all of that.
I think people create a false dichotomy when dating again after divorce when children are involved. They start from the premise that they have to choose either their or their children's desires. I suggest that is the wrong premise and there really is no dichotomy at all because there is no "choice it is not about choice. Your children are a part of you and any woman who loves you will love them or she will not really love you. That said, there are practical realities and, as 25yearsmlc points out, she should probably not even be introduced until you are sure enough (whatever percentage works for you) she is someone with whom you want to build something.
I think your kids will be a great guide for you on this path. They will sense if you are loved in a way they want for you. All of you will feel that. It is for this reason I say there is no "choice" to be made.
Wow, that's a lot of advice from a woman without children, but I am someone who will be a part of someone else's life one day and this is my approach. It really is about loving the whole man and if that man has children, then his children are a part of that whole man.