I associate
away with continuity of action and a certain amount
of absorption of the mind in the activity, mixed in many cases with an
enjoyment of the activity. It's not a particularly formal device.
She was happily -- singing away; typing away; cooking away; whistling away
It doesn't work with every verb, of course,
The sound of the music was dying away in the distance.
and it cannot be used in this meaning with statives,
*living away, *sitting away, *depending away, *resting away
and with many verbs, especially, but not exclusively, with verbs of motion, it has its literal meaning:
run away; walk away; go away
keep away; stay away; be away
CJ