I am not saying is heard often. It is not a complete idea. It needs to be followed by what is not being said. It clarifies what the speaker sees as a possible miscommunication. If I think you may believe I am saying one thing when I am not, I may want t clear it up for you. What is not being said, sometimes is being hinted at... "I am not saying I expect to be paid for this explanation, but you could choose to do that."

Kinda. A good example of the "be doing smth" being something more, semantically.