I'm Dr. Jackie, an interpersonal communication researcher/professor and relationship expert. And first, congrats on your marriage! Second, I am sorry you are going through this conflict. Perhaps I can give an objective view on this; and hopefully, your new husband will be willing to read my response and reflect on this also.
The most disconcerting part to me about the entire post is that your husband says that she is his only true friend and vice versa. This saddens me, and I hope it is not true--You and your husband should be best friends. If he truly feels that way about her, then I do believe emotionally he certainly has "cheated" on you. And while he may not have romantic feelings toward her at all but friendship only, I'm fairly certain this is not the case for his ex-girlfriend. Her behaviors and attitude definitely seem to reflect differently.
When your husband took his wedding vows, he promised to put you above all others, and this wedding vow does not make an exception for an ex-girlfriend. So the fact that he did cut the contact considerably is definitely a move in the right direction. But if this is still causing friction between the two of you, his loyalty needs to be to YOU. He married YOU. So you and your feelings need to come before hers. If you don't get this resolved now, it is going to continue to haunt you like it is by keeping you awake at night.
I do want to point out something, and I want you to understand I will always be forthright and direct and tell you my opinion based on experience with doing this for a long time as well as based on what research studies reveal. And so I do want to mention something, and I hope it will make sense. You say that, "He makes [you] feel like the crazy woman..." Think about the wording if you will. Short of extreme examples where someone truly is forced (by real threat), in most instances, we must take ownership of our feelings. No one else can really MAKE us feel something. In fact, we can't even control our feelings. They are emotions inside us. What we CAN control is our REACTIONS to our feelings and REACTIONS to others' actions. I think if you think about this notion, it truly is empowering that while you can't control feelings, you DO have the power to control how you react. And exercising self-control and gaining control over feeling resentment toward this woman should be somewhat empowering to you. That may be helpful. Obviously, I am not dismissing your husband's role in putting you and your feelings above the ex. That remains very important as well.
Finally, talk to your husband. Be honest with him about how this is really hard to get past. The more open and direct you are, at least you are not "pretending," and it shows you trust him that you are sharing it. Most often when one person in a relationship is very honest and direct and it shows, the other person also tends to be more direct and open. This is one way to prevent any secret texting, calls, etc.
What do you think so far? Would he be willing to talk to a third party (ex: counselor) about this? Everyone can use some good guidance and counseling at some point and time. And if you would like to discuss this via the phone or Skype, I'll send you the JUST ANSWER form with my information on it so that you have it.
Please take care. I hope to hear back soon.