Are there any legal or financial consequences to entering into a new relationship while still married?

My husband and I separated (not legally) in 01/2014. I moved out of the house, we finalized custody and child support in family court, but divorce papers have not been filed. I want to enter into a new relationship, but am afraid that doing so might entitle my husband to some argument for money somehow. I have little understanding of legalities, but it's my understanding that adultery is illegal in New York. If he wanted to, and could prove such, could there be legal or financial consequences for me? Would I have to worry about consequences related to custody of my child?