Does the statement of the two Sheikhs, Ibn Uthaymeen and Ibn Baz, regarding not paying attention to doubt if it does not reach the degree of certainty, apply to doubt about uttering a vow, especially if the side of having uttered it is more likely?

The basic principle is that whoever doubts whether they uttered a vow or not, then they have not uttered it. One does not depart from this principle except with certain conviction. One is not judged to have made a vow except with certain conviction, whether it is highly probable or not, because the original state is freedom from liability, and this is certain, so one does not depart from it except with certainty. Among the proofs for this is the Hadith (definition) of the Prophet, peace and blessings be upon him: "He should not leave until he hears a sound or perceives a smell."

And one must beware of obsessive thoughts (waswas), for indulging in them leads to great evil.