What is the lawful way to get to know a girl, whom one believes to be righteous, with the intention of marrying her, while avoiding embarrassing her or wronging her family?

It is advisable that you ask one of your mahrams (unmarriageable kin), such as your mother or sister, to speak with the girl to get to know her and her family, and to inquire about her. If she is found to be righteous, then propose to her family. If that is not possible, then seek an elderly woman to do so. If all of the above is not feasible, then there is no objection to you speaking with the girl directly and asking her, while observing the Islamic guidelines. Then, if marriage to her is not facilitated, seek out other righteous women.