What is the difference in what your cousin is suggesting and pursuing/pressuring the WAW?
Why would you tell her that you will keep your light on in case she decides to come back?
And are you going to support her if she's having an A?
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but in my humble opinion it might make all the difference to her to actually know how you really feel right now as opposed to hearing nothing from you and possibly reaching her own self-fulfilling prophecy confirmations like, "See, the fact that he's not saying anything only proves that he doesn't care or isn't interested in me."
This is not the thinking or reaction of a WAW who is in an A! And, if you found inappropriate contents between her and OM, then she's probably having an EA, if not a PA by now. If that's the case, you don't need to appear to be available (leaving the light on) b/c it just pushes her away.
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It will also show her that you care enough about the relationship to invest time in counseling, reading, and self reflection, and you are willing to make changes for the future. It's possible that these might be sticking points for her. Her knowing how you are currently making changes might be the start of the "heat" she's looking for to melt away the ice that has formed between you
Again, how is this different from any other pleading & promise making? A WAW doesn't want to hear how much you love her (or care about the M)and what you're willing to do to keep her. She doesn't want you convincing her how much you care! If she had left under different circumstances, there might be an exception, but based on your written post, I just don't agree with your cousin's advice. We always tell the LBH to improve himself, but not to talk about it to his W. If you really change, she'll hear about it, or see it for herself.
It is not about what you feel should work in your M. It is about doing the work that gets the right results. Do what works!