Obsessive leading lady departs in state (5)
I believe the answer is:
nerdy
'obsessive' is the definition.
(as in being nerdy about something)
'leading lady departs in state' is the wordplay.
'leading lady' becomes 'ER' (Elizabeth Regina).
'departs' becomes 'd' (common abbreviation e.g. train timetables).
'in' means one lot of letters goes inside another.
'state' becomes 'ny' (short for New York).
'er'+'d'='erd'
'erd' going into 'ny' is 'NERDY'.
(Other definitions for nerdy that I've seen before include "Bright but socially inept" , "Boringly studious" , "obsessive" , "Geeky" .)