The change can be described as either a change in frequency with time, or of phase with time, or both if you really want to make it complicated.  There is also a change in amplitude with time.  This can be analysed using complex numbers, or real numbers.  I shan't try to explain how to do it with complex numbers.

In real numbers, plot the peak levels as a function of time and get the (probably exponential) function exp(-at) that describes the reduction in amplitude.  Then multiply the signal by exp(at) (or divide the signal by the function you get if it isn't exponential) to take out the change of amplitude.  Now compare the signal at every point with a sine wave of the same amplitude and similar period (or with the other decaying signal treated in the same way to take out the decay), measuring the time interval between points of the same amplitude on the two curves, perhaps at 10 places per period.  Plot the time interval against time and if it is a straight line the frequency is not changing, and the slope can be used to get the exact frequency difference between the two sine waves.  If the line is curved, then the frequency is changing and the phase change with time p(t) can be obtained from the equation of the curve, perhaps using a polynomial fit - the linear term in time can be used to get the frequency difference between the mean frequencies of the two signals, and the quadratic term to get the linear chirp, and so on.

If you are working in analogue and have a reference signal at something like the same frequency, you can put the signal and the reference  both into an quadrature mixer, and put the two outputs into the x y channels of an oscilloscope.  If the trace is a straight line from the edge to the middle then the reference signal is at the same frequency for all the signal.  If it is curved at any point, then the rate of rotation round the centre with frequency can be used to get the frequency difference for that part of the curve.  Having a  variable frequency reference would be useful at this point.  When the curve is pointing at the centre (is tangent to a radius) then the tangent point gives the amplitude of the signal when it is at that frequency.  You can also compare the phase of two decaying signals in the same way by putting them into a quadrature mixer.

This task has classical solution in FM-radio reception: Amplification with saturation.  The quasi-sinusoidal signal transforms to rectangular.

I would initially use some method of morphological curve fitting. It seems to be the only way to avoid natural phase distortions inherent to linear filters, for short signals. After this preprocessing, which is a nonlinear bandpass filter without introducing distortion in the phenomenological underlying physics, the use of phase morphological parameters as a form of measurement is straightforward.

For each individual signal, the 'local' peaks (min and max) are easily calculated. The differences give the frequency (if changing at all) and the oosition of the min and max values can be used as well to derive the phase relationship.

This may sound unconventional, but if you do not know the exact zero-level, this is a method which does not require this knowledge.

On the other hand, if the zero-level is well known, you can use the zero-crossings to calculate frequency and phase relationships.

Regards

I think you could apply a FTT to the signals and then calculate the angle between them. 

regards,