Act indecently? I become tight-lipped (8)
I believe the answer is:
taciturn
'tight-lipped' is the definition.
(taciturn means not saying much)
'act indecently? i become' is the wordplay.
'indecently?' indicates anagramming the letters (I've seen 'indecently' mean this (I've seen this in other clues)).
'become' becomes 'turn' (synonyms).
'act' with letters rearranged gives 'tac'.
'tac'+'i'+'turn'='TACITURN'
(Other definitions for taciturn that I've seen before include "Reticent" , "Without saying much" , "Uncommunicative, of few words" , "Habitually silent" , "withdrawn" .)