What is the difference between sampling over $-4\pi$ to $4\pi$ and 0 to $8\pi$
Your spectrum is zero at all frequencies other than at the carrier and the side bands. If the magnitude is zero the phase is undefined. Since you use finite precision math, the magnitude in your calculation will not be exactly zero but just a really really small number. So the phase can be computed but what you are seeing just random numerical noise.
As you change the time interval, the noise will change as well, but the magnitude will still be extremely small.