It’s very hard for me to assess a situation like this on the basis of a letter. It might be that your husband is more attracted to this woman than he is admitting to you or to himself. But the fact is, he’s been very open with you and has not acted defensive in any way – even when you asked to see his texts. That suggests that at the very least he’s trying to be respectful of your feelings.
The other possibility that occurs to me is that your husband is lonely for a friend. Working night shift makes it tough to have friends or even to be as close to you as he might want to be. Then along comes this person who he not only likes but who shares the same difficult schedule. It’s possible that what you are sensing is the beginning of a romance. But it’s also possible that what you are seeing is the enjoyment of a new friend.
If you two were seeing me in my office, I would want to ask you if you each have close friends you do things with and confide in. For many people, having young children and working third shift tend to crowd out having friends. Our partners can’t be everything to us. People need a social group. If your husband doesn’t have friends, I would then suggest to you that his solution is a healthy one. Make friends with her too and expand your social circle.
On the other hand, if your husband does have good friends in his life, I would wonder why this particular relationship is so important that he is willing to hurt your feelings to continue it. Is he feeling sorry for a single mom? Or is he letting himself get more involved with her than he should because it is kind of exciting to be around her? The answers to those kinds of questions would help you both figure out what role this woman has in his life and if it is a threat to your marriage.
I’m sorry to give you such an unsatisfying answer to your question. That always happens when an answer is “it depends.” What I do suggest you do is keep talking. Perhaps some of the ideas I’ve offered give you a different direction for your conversation. Work hard not to accuse or blame. Instead, just try to be interested in what it is that he is getting out of the relationship. Once you understand that, you’ll be in a much better position to make your own decisions.
I wish you well.