What is the responsibility of a person whose Spouse pays child support?

My husband and I are registered as Domestic Partners and NOT husband and wife. The only reason we have not married through the state of New York is because he has hefty child support responsibilities with 2 children he had with his ex. We had a ceremony but never filed the papers because he is afraid that his ex would be entitled to a portion of MY earnings as well if we did. Something about total household income increasing. Is this true? Should we wait until he stops paying to marry? Is there a way around the support getting increased, like if we file seperate taxes? How about if we just marry and file but don’t tell anyone? Would they still find out? It really bugs him that we aren’t legally recognized as married and that it’s not on paper.