Igros Moshe OC I:113 and EH I:56
The Steipler writes in his letters in the name of the Chazon Ish that it is absolutely prohibited.
Regarding a so-called heter that you will embarass her if you don’t shake hands, it’s simple logic: If someone asked you to call them on Shabbos and you telling them sorry I can’t do that would embarrass them for asking, would you be mechalel shabbos to prevent their embarrassment? Or if someone offered you non-kosher food and refusing would embarrass them would you eat treif?