I'm so sorry for your troubles regarding these domestic issues with the kids' father. Custody/visitation and all of that can cause so many disruptions, and stress. Try to hang in there!
Unfortunately, you likely can not prevent him from venting that he will take an issue back to court, if he feels that things are not going properly, or if he feels that you are not abiding by Divorce/Settlement agreement (even if you are, if he does not perceive it that way). If you know you are abiding by your obligations, I recommend just ignoring his comments, and they may just stop. Going to court is a bit of a hastle and he probably would prefer not to. If you are not abiding by your court ordered obligations (perhaps, you feel, justifiable not, or you are actually unable to do so), you may want to consider going back to court to get try to get those obligations reducesd or changed to be more workable - this would prevent you from being in contempt of the existing court order.
From your statement, it appears that you had his name put on your eldest child's birth certificate, is that correct? If this involved an under oath statement that you knew to be false, please be careful with regard to this issue, so as not to cause more trouble for yourself.
If you were to pass away (gosh forbid!), their father, being their only parent, would likely succeed in getting full custody. He is the father and short of him being deemed an unfit father, the step-father would take a second seat.
Hang in there! I know your children may be young and so it seems this ongoing unpleasant interaction with the father may go on and on and on, there is a light to the end of the tunnel when they become of age. If you can be as amicable as possible and let the insignificant (even if unfair) things roll off your shoulders, it may be easier to deal with until you legally don't have to anymore. Here is to that day!
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Stephanie Joy, Esq.