Living without sin's first dying (7)
I believe the answer is:
exiting
'dying' is the definition.
('exit' can be a synonym of 'die')
'living without sin's first' is the wordplay.
'living' becomes 'existing' ('exist' can be a synonym of 'live').
'without' indicates named letters should be taken away.
'sin's first' becomes 's' (1st letter of 'sin').
'existing' with 's' removed is 'EXITING'.
(Other definitions for exiting that I've seen before include "Going out" , "Leaving" .)