I am a lawyer in CT and practice in this area of the law.  Unfortunately, you may not sue your husband.  most state statutes do not permit spouses to maintain actions against the other for adulter, especially in this situation where you are separated awaiting to file the divorce papers.  Nothing precludes your husband or you from dating other people.

First of all, without proof, your divorce will in no way be affected my your husband's alleged affair.  However, if you are able to get proof, that would change things greatly.

In North Carolina, you are free to associate with whomever you choose prior your divorce, but until a final decree of divorce is entered, you are still married.  It is important insofar as there has been marital misconduct during the marriage.  In that case, the granting and amount of an alimony award, can be affected. 

Additionally, you may be able to sue your spouse's lover if certain legal requirements have been met. 

Alienation of Affection is a legal action based on willful and malicious interference with marriage relations by a third party.  In a divorce matter, Alienation of Affection actions are often brought along with Criminal Conversations actions.  For a plaintiff spouse to recover for Alienation of Affection, the following elements are required: (1) the parties to the marriage were happily married and genuine love and affection existed between them; (2) such love and affection was alienated and destroyed; (3) and the wrongful and malicious acts of the defendant brought about the loss and alienation of such love and affection.  The exclusive right of sexual intercourse is not the right protected in this type of case.  The actual affection between spouses is the right protected.

Criminal Conversation is a claim for adultery.  Unlike alienation of affection, this action does protect the fundamental right of exclusive sexual intercourse between spouses.  To prove a cause of action of criminal conversation, the plaintiff must be, or have been at the time, lawfully married and his or her spouse had sexual intercourse with the defendant without the consent of the plaintiff during the marriage.  It does not matter if the plaintiff and his or her spouse were separated at the time.