Here's some of my experience about urges. Giving in, for me, made the "noise" in my head disappear for a while only to return with a much greater vengeance later on. Shame, guilt, etc. was temporarily deleted from my being by build up part of the cycle to the actual sexual acting out and then after the negative feelings would come back. It took me a long time to identify this pattern.

For me, there have been two most helpful things. My willingness to be honest about my urges and what I did whether fighting them or "giving in" to them and then sharing that honesty with someone that I trusted. I think I read that you are looking for a therapist. I found one and then another and I talked and talked and talked about this stuff and I found my confusion eased. I talked about when I gave in. I talked about when I was fighting it. I talked about what I thought I should be thinking about. I talked about things I didn't want to talk about.

I have found my confusion eased, I have found greater acceptance of myself and my feelings and much less anger at myself for having any urges. This post is not to suggest any course of action for you or anyone else. It is just my experience.

"Does understanding them lead to acceptance?" I guess it was a combination of both. Accepting them led to understanding them and vice versa. I did this in pieces and my goal was greater understanding and internal peace and I have more of both. You seem like you are seeking that and I believe you will find it if you make whatever efforts you deem the course for yourself.
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And more, much more, the heart may feel,
Than the pen may write or the lip reveal.
Winthrop Mackworth Praed