Running over those characters starting for all sides today (4)
I believe the answer is:
full
'running over' is the definition.
(thesaurus)
'those characters starting for all sides today' is the wordplay.
I cannot quite see how this works, but
'sides' could be 'll' (an abbreviation for left written twice) and 'll' is found in the answer.
This explanation may well be incorrect...
Can you help me to learn more?
(Other definitions for full that I've seen before include "Treat cloth; complete" , "Complete; replete" , "At maximum capacity" , "Having no empty space" , "Able to eat no more" .)