Question

True/False: If an object stays still, but the observer moves, the Doppler Effect is still observered

Answer 1

The answer is to your question is true....plain and simple

Answer 2

Well, Harry, what you said is not necessarily true the way you said it. 
But we know what you mean, and what you meant to say is true.

The Doppler effect is observed if there is relative radial motion
between an object and an observer AND if the object happens
to be putting out sound or light in the observer's direction.