You may feel like your marriage is over, even before you file your divorce paperwork, but legal experts generally advise against living with someone else until after your divorce is final. New relationships during divorce can complicate the divorce process by creating emotional conflicts and can even affect spousal support awards.
You may have moved on emotionally, but it is possible that your spouse hasn't. If you commit adultery, your new relationship may be legally relevant to your divorce. If your current spouse wants to drag your new relationship into the courtroom, depending on the laws of your state, he may do so. This can make your divorce emotionally draining for you and your new partner, and it can extend and complicate the divorce process.
Living with someone can have financial consequences, in some cases. You could see your spousal support possibilities limited or even lost, because of your new relationship. Your divorce court may look at your new living situation and decide that you don't need spousal support, since you are able to share living expenses with someone else. If you are spending marital assets such as money from a joint bank account on your new partner, the court could also reduce your share of the property split in the divorce, since you wasted marital funds on an adulterous relationship.