Sawdust is used in many places at absorb spilt liquids such as blood, vomit and urine in butchers shops, abattoirs, circuses etc.
The former soldier has killed both on the battlefield and states "I have killed on the sawdust" I would interpret this to mean he has killed in an arena as entertainment as a form of gladiator, not having read the book. The floor of the arena would be covered with sawdust to absorb the blood etc. split by the dead.
It occurs to me know it could also mean he has killed on the training room floor, in some extremely violent warrior societies occasionally killing real people was part of the gaining. Spartans would have to kill a helot slave before they were considered real soldiers for example. Sawdust would act as an absorbant there as well.