Not sure about the laws in your state, but I know that Deb had a provision in her settlement agreement that stated that her ex could not have the children overnight if he had his female interest staying overnight as well. For that reason he picked up and dropped off his children every saturday and sunday for several months.
I have insisted with my ex that she not sleep with the man she is living with whenever the boys are visiting. To the best of my knowledge she has honored that request.
Depending upon your belief system, this can be a very significant issue, but it's one you have to pursue with your Lawyer. Only he/she will be able to tell you if such a provision is possible.
And I will say that I think it's intolerable to co-habitate with your children present. We are responsible to teach our children something about what appropriate behavior is and is not.
Bill
"Don't tell me the sky is the limit when there are footprints on the moon."