One's been chilly -- I stay, oddly, showing foolishness (10)
I believe the answer is:
imbecility
'foolishness' is the definition.
(synonyms)
'one's been chilly i stay oddly' is the wordplay.
'one's' becomes 'im' ('one is' is a formal way of saying 'I am').
'oddly' indicates alternate letters (letters in the odd-numbered positions are used).
'been'+'chilly'+'i'+'stay'='beenchillyistay'
The alternate letters of 'beenchillyistay' are 'becility'.
'im'+'becility'='IMBECILITY'
'showing' is the link.
(Other definitions for imbecility that I've seen before include "It's sheer stupidity!" , "Stupid act" , "folly" .)