Extreme sexless marriage.. young couple.. please help

I am 36, she is 39. We dated for 3 years, and have been married for 11. After the first few months of dating, the sex dwindled to a couple times a month. By the time we were married, it was already pretty non-existent. I know, I know, "Well then why did you get married?" When I discussed this giant red flag with my wife,(fiance' at the time) she said "did you ever stop and think that maybe I just want to save some things for marriage?" I respected that, and believed it. Imagine my heartbreak when we didn't even have sex on our honeymoon. We have never had sex on an anniversary, birthday, valentines day, holiday etc. I stopped persuing it, as it just lead to heartache, and fights. I thought maybe I could just live without sex. I actually made a conscious decision to just stop bringing it up, and surely, she would see how this was hurting me and then she would take steps to fix it. That plan backfired, as we went completely, 100% sexless for five years.I never once cheated, although I was tempted often. She told me that she is just not a sexual person, and has no sex drive. I suggested on multiple occasions that she go to a dr. to have hormone levels checked, as this could be a very easy fix. She flat out refuses to go to a Dr for this issue. I have suggested counseling/therapy. She flat out refuses that as well. She is not getting it (sex) anywhere else. We spend nearly all our time together. We really do love each other very much. I would take a bullet for her. I asked her if she was just not in to me, and wanted a divorce. She says no. She loves to snuggle, and be held, and be close, just no sex. I'm at the end of my rope. An affair, although tempting at this point, would require some sort of emotional connection with the other woman, and I just imagine I would feel so guilty. I am now contemplating, as crazy as it seems, visiting an escort a couple times a month just to satisfy my sexual needs that my wife refuses to address. When I imagine doing this, I feel no guilt. It's a woman I don't know, and there would be no emotional connection. I almost feel that in my extreme case, I could justify it. I just want the excitement of sex, and the touch of a woman. Divorce is not an option. I love my wife too much to not be with her anymore. It's just this one GIANT issue that will not go away. Am I crazy, or even a "dirt-bag" for considering going this route? Why do I feel like if I hire an escort, in my situation, it would not be cheating?