I think you’d be best tying either Amplitude or Frequency of the wiggle to a slider, then keyframing the slider. Zero as a keyframes kills the wiggle dead, a positive value down the line builds to the wiggle gradually. Hold keyfraems make for an abrupt start/stop for wiggle.
In case you don’t know how:
wiggle (5,20)
highlight (select) the 5, then pickwhip to a slider stopwatch(Effect>Expression Controls>Slider Control) and you have an expression like this:
wiggle (effect(“Slider Control”)(“Slider”),20)
Setting keyframes with the slider now puts you in complete control.