I live in NY. My children's father hasn't seen our daughter (who is almost 2) since June, and hasn't seen our son(who is almost 3 months) at all since he was born January 2nd. He doesn't try to get a hold of me to see them, ask how they are doing, ask if they need anything, or offer any help. He doesn't pay child support or help me in any way. He inst on either one of their birth certificates, but there is nothing established in court that I have full custody. Is this abandonment? If not what is it and what should I do?

Have you filed to establish a support order? (I'm assuming No, since he is not listed as the father on a BC for either child) Support isn't legally enforceable without a support order. That should be your first step. You cannot force him to visit or to be a good parent, but he can (and should be) financially responsible for his children.

No I haven't taken him for child support because the moment I do that I have a feeling he will try and take me to court for some kind of visitation or custody just to be a jerk because I took him for support. I don't trust him, and don't want myh children around him.

You not asking him for support does not stop him from getting visitation if he wants it. He is the father, and unless he is incredibly unfit, he will get at least supervised visitation. That isn't related to child support at all, and he can as the courts for visits at any time, regardless of his payment (or lack thereof). Certainly you knew he was a jerk whom you didn't trust before the second child.

He can not abandoned a child that is not legally his. You have never established that he is the father. He is not on the birth certificates and he has not been proved the father by DNA testing.

Next of course he does not pay child support, there is no child support order in place.

Child support and visits are not connected, he has rights to visits without paying support.

Next in one line you fuss about him not visiting the kids, but then you don't want to ask for child support since he may "want" visits. You appear to want to control him somehow to visit you more than the kids, or have money without him seeing the kids.
Sorry it does not work like that.