How can a young man become a son through rada'ah (suckling) or any other legitimate Sharia-compliant way to this woman who considers him as her son, so that she can greet him without embarrassment, knowing that adoption is forbidden in Islam? And can she be considered among those "without carnal desire" (ghayr uli al-irbah) at the age of 42?

There is no way to make you a son to this woman. She is not an old woman nearing death for you to be lenient in dealing with her regarding looking at her or shaking her hand. Rather, she is a stranger to you, and you must observe the Sharia limits in dealing with her. It is not permissible to be complacent regarding these limits. However, if you marry her daughter, she becomes a mahram to you, and what is permissible with mahrams regarding looking and shaking hands becomes permissible with her.