Does swallowing phlegm that has reached the mouth invalidate the prayer, and is my prayer valid in this case?

What you heard about the invalidation of fasting and the nullification of prayer by swallowing saliva is not absolute.

Accumulating saliva during prayer is a futile act and is not permissible.

Rushing in prayer invalidates it if it compromises the element of tranquility; otherwise, the prayer is valid.

The cure for obsessive thoughts is to disregard them and pay no attention to them.