Is what the questioner feels when reading Prophetic hadiths—a sense of disbelief and dislike, despite trying to convince oneself with arguments—a form of waswasah (whispers/obsessive thoughts) or doubt? And is the saying "faith is the faith of old women" correct? And is it true that uttering the two declarations of faith helps one remember if they forget, and is this found in religion?

What you are suffering from is a whisper (waswasah), and its cure is to completely ignore it, pay no attention to it, seek refuge with Allah, and stop engaging with it, as those who were successful have experienced. Indeed, persisting with it only increases it until it drives a person to the brink of madness.

As for uttering the two Shahadas to remember something when forgetting, there is no evidence for it. However, it is recommended to engage in dhikr (remembrance of Allah) when forgetting, based on the Almighty's saying: "And remember your Lord when you forget" (Al-Kahf: 24). Some scholars interpreted this to mean remembering the exception "in sha' Allah" (if Allah wills) if one forgets it when speaking about the future, while others interpreted it to mean remembering Allah generally when forgetting, to drive away Satan.

When Sheikh Al-Uthaymeen was asked about dhikr when forgetting, he replied that asking Allah to remind one of what they have forgotten, by saying, for example, "O Allah, remind me of what I have forgotten," is more appropriate.