Numerical Calculation More Stable with Higher CFL number?
 

Thus, I planned to use the Lax-Wendroff technique for numerical linear advection analysis. In terms of code, I believe my code works as planned. Nevertheless, it appears that my CFL=1 result is more stable than CFL = 0.5 when I run the code with different CFL values. Does it make sense? Or is there a mistake in my code?

No. The L-W is stable for c<=1. The specific case c=1 produces the exact solution but for c<1 the solution is always stable. What you see is the face that the solution is not monotone.
These are issues you should study in a CFD textbook.