My children's birth certicficate has only my name on it, so does that mean I have full custody of them? If not how do I go about making sure I have full sole custody?

Unless your boyfriend comes along claiming the kid and wanting to fight for his parental rights you would be the only one with full custody.

If threre is no name on the birth certificate and if you were not married when you had the children at this point you have sole custody, but the biological father can request a DNA test and seek joint custody or visits and other rights if they want.

Until someone challenges your for it in court... you have full/sole custody.