What is the ruling on a young man secretly becoming engaged to a girl other than the one his family desires for him, with the aim of preserving his chastity before traveling, and is he sinful for doing so?

You must obey your parents and treat them kindly, as long as they do not command a sin. If your cousin is religious and beautiful, and they command you to marry her, then you must obey them. Your reluctance might be a whisper from Satan. Allah may place much good in this marriage. However, if your cousin is not committed, then seek a religious woman after consulting your parents. If they permit you to marry whom you wish, then that is excellent. Otherwise, abandon the matter and obey them.