This sounds like a classic negative feedback oscillation. You sense the
signal is too large, so you send a signal to kill the gain, and then
you sense the signal is too small, so you send a signal to increase the
gain.

Having different attack and release time means you have two different
time constants My guess is the quick attack leads to the instability,
since it is the lesser damped system. If this is true, then you should
concentrate on the attack time, i.e find how slow it has to be for the
sytem to be stable.

Of course this is really had to do without seeing the circuitry in
action.