Terribly serious following new obsession (8)
I believe the answer is:
neurosis
'obsession' is the definition.
(I've seen this before)
'terribly serious following new' is the wordplay.
'terribly' indicates an anagram.
'following' is a charade indicator (letters next to each other).
'new' becomes 'n'.
'serious' is an anagram of 'eurosis'.
'eurosis' after 'n' is 'NEUROSIS'.
(Other definitions for neurosis that I've seen before include "Trouble in mind" , "Mental disorder of resinous origin" , "Mild mental disorder - resinous (anag)" , "Mild mental illness not caused by an organic disease" , "Mild condition such as obsessive behaviour" .)