What is the ruling on saying "we will part ways" or "we will divorce" falsely and without the intention of divorce, with the intention of clarifying the gravity of the problem to the mother out of anger and with the presence of whispering (waswasah) in the statement?

Divorce does not take effect by falsely reporting it. If you explicitly uttered it due to overwhelming waswas (obsessive whispering), your divorce does not take effect. If you doubt whether you uttered the divorce, or what you intended when you uttered its كناية (allusion), then nothing takes effect, because the original state is the continuation of the marriage.