What is the ruling on an unmarried young man continuing to communicate with a married woman—even if the conversation is religious—given that they have become attached to each other, and are both of them sinful for this?

Your relationship with this woman is forbidden (haram), and corresponding or chatting with her after repentance is a clear abomination and proof of your insincerity in that repentance. You must sever your relationship with her immediately. Your discussions with her about religious matters and your cooperation in good deeds are a deception from Satan. Hasten to protect your chastity through marriage, for it is a fortress for you.