Old kingdom is under obligation to drop first one, then the other (6)
I believe the answer is:
wessex
'old kingdom' is the definition.
(historical English kingdom)
'is under obligation to drop first one then the other' is the wordplay.
'is under obligation' becomes 'owes' (to owe is to be under obligation).
'to drop' indicates named letters should be taken away.
'first one' becomes 'o' (1st letter of 'one').
'then' says to put letters next to each other.
'the other' becomes 'sex' (old-fashioned slang term).
'owes' with 'o' taken out is 'wes'.
'wes'+'sex'='WESSEX'
(Other definitions for wessex that I've seen before include "ancient region" , "Alfred the Great's realm" , "Anglo-Saxon kingdom" , "Hardy's West Country" , "South-west counties of Hardy's novels" .)