Cross correlation vs FFT for finding phase between 2 signals I have been struggling with this issue for months now, trying to understand which one to pick: FFT or cross correlation. 

I am given a sine wave (with phase, amplitude, frequency and time vector known), and I have to extract the phase shift between that sine wave and another signal. The new signal has the same frequency and the same time vector as the sine wave, however the amplitude and the phase are unknowns. What would you guys suggest? I am at a loss.

Thank you.
I don't see it as one or the other. You can use the FFT to more efficiently compute a cross-correlation (versus computing the cross-correlation in the time-domain). 

There is lots of good information on this thread, about computing the "phase shift", from the offset of the peak in the cross-correlation function.