Does seeking ways to eliminate self-whispers (waswas) constitute speaking or acting, thus invalidating the hadith: "Indeed, Allah has pardoned my Ummah for what they whisper to themselves, as long as they do not act upon it or speak of it"? And how can a person ignore these whispers, repel them, and despise them at the same time?

Your question is somewhat vague, and the ambiguous talk it encompasses is the self-talk that is forgiven. The servant is not held accountable for this talk merely because he read about how to get rid of it, because this reading is not considered speaking about it or acting upon it. There is no contradiction between a person disliking it and resisting it, and ignoring it. In fact, ignoring it is considered part of resisting it. So, he distracts himself from it with something else that benefits him and profits him in his religion or worldly life, and this is proof of his dislike for those whispers.