How to recover frequency and damping coefficient from superposition of damped oscillators?
If you are absolutely sure that you have a signal that is the superposition of damped oscillators then you can simply track their evolution in time.

Fs = 2000; % Sampling frequency (Hz)
T = 4;     % Time duration (seconds) 
t = 0:(1./Fs):(T-(1./Fs)); % Time vector (in seconds)
p = 2.0.*pi.*t; % Phase vector
S = 1.0*sin(440.*p).*exp(-t) + 0.25 .* sin(800.*p).*exp(-t*10); % Signal


This signal looks like:



And via a Short Time Fourier Transform (STFT) (surf(stft(S))) it looks like this:



Granted, if you have beating frequencies, the task is getting a bit more challenging. For instance:

S = 1.0*sin(440.*p).*exp(-t) + 0.25 .* sin(442.*p).*exp(-t*2) + sin(800.*p).*exp(-t*10);

Would show up as:



But once you see oscillations on the damped decay, you know you have beating and you can recover how close the beating pair is by measuring the beating frequency and how quickly it fades (to derive its decay).

In a similar way (i.e. using the STFT) you can also recover the phases of the sinusoids.

Hope this helps.

UPDATE:

The STFT of the signal that was shared via pastebin is capable of distinguishing three components in this signal. This is as follows:

%Given a signal in s and that the signal package is installed (in Octave)
[S,F,T] = specgram(s);
surf(abs(S));


This returns: