Permission for adultery?

Do not put anything in writing until you are formally separated and have a legal separation agreement. Although you can agree to the terms without an attorney and “do” a divorce yourself, I would not do anything that may damage the case. Until you are divorced, you are still married. It may be that he is already involved with someone. If that is the situation, and you are the dependent spouse, he owes you alimony. Your permission for him to “date” others will nullify that claim.

Technically you cannot give your husband permission to commit adultery, which is still a crime in North Carolina. If he wants you to put it in writing that he can date other women it will not hurt you if you do not plan to pursue alimony.
Even if you did sign a document allowing your husband to date other women, it would not bar your alimony claim, it would give your husband a defense to an allegation of adultery, which guarantees and alimony award for a dependant spouse. Condonation does not mean that you would not receive alimony if you can demonstrate you are a dependant spouse.
In relation to custody, allowing your husband to bring other women into his life while the two of you are still living together is not something that is likely in your child