Does swallowing one's saliva, while feeling its coolness and experiencing obsessive doubts about the presence of specks or dots in the mouth after rinsing and spitting, invalidate the fast?

The cure for obsessive whispers (waswas) is to turn away from them and not pay them any attention. These whispers must be ignored and not given any importance. Fasting is not invalidated by swallowing saliva, even if it is mixed with moisture from the effect of rinsing the mouth.