Offend royal that's boring? (7)
I believe the answer is:
sinking
'boring?' is the definition.
(sink can mean to drill or bore down into earth)
'offend royal' is the wordplay.
'offend' becomes 'sin' (sin can mean to commit an offence).
'royal' becomes 'king' (a king is royal).
'sin'+'king'='SINKING'
'that's' is the link.
(Other definitions for sinking that I've seen before include "Falling to the bottom" , "Going down" , "failing" , "in decline" , "Declining" .)