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LW,
I would not presume to offer up an opinion about whether your husband has cheated with this woman. What I do know is that in almost every relationship I've ever been in, whenever I've practiced some form of avoidance in talking with my significant other, I was hiding something. It wasn't cheating, but it was always something I didn't want her to know for one reason or another. Sometimes it was just an emotion I felt uncomfortable revealing. In the best relationship I've ever had, my S.O. wouldn't stand for it, and worked hard to break down that barrier. I fought it. I don't think I ever said she was crazy or 'got like that', but I did insist she stop bugging me. The bottom line is that she wouldn't take no for an answer, and eventually I stopped evading her. I made everything in my life transparent to her, and she in turn, felt safe and stopped 'bugging me'. The closeness and trust that ensued was priceless.
You may or may not have a cheating problem. I have no idea. But you are not sure if you trust him, or you wouldn't have written the letter, and he doesn't trust you enough to open up to you (or, he is, in fact cheating). Even in the absence of cheating this lack of trust is toxic to your best relationship. If I were him (and I were not cheating), I'd thank you later if you didn't take no for an answer.