Does divorce occur upon the statement "that each goes their own way," with the intention of going to court for divorce proceedings and not intending an immediate divorce, especially since the wife is pregnant and the husband is going through psychological distress?

This phrase is not an explicit divorce pronouncement; rather, it is an ambiguous expression (kinayah) that can signify divorce or something else. As long as there was no intention to enact divorce immediately with it, but rather an intention to divorce in the future, then no divorce occurs as a result of it. An ambiguous expression does not effect a divorce without intention.