How to detect decaying oscillations in a signal
Your signal model lends itself particularly well for so called parametric frequency analysis. The basic idea is that you find the autocovariance matrix of your signal and identify its eigenstructure. The eigenvalues correspond to the exponents of a sum-of-exponentials model. In your case you will find conjugate pairs of eigenvalues with non-zero imaginary part describing the exponential decay.

This thesis looks like a good introduction to the topic:
http://zet10.ipee.pwr.wroc.pl/record/11/files/Leon_mon.pdf