Yes, the ships have self serve launderettes.
Look here for some more information about the prices etc.
On the Carnival Legend they are located as follows;
Riviera Deck 1 midship across from 1219
Main Deck 4 aft across from 4230
Upper Deck 5 aft across from 5270
Empress Deck 6 forward across from 6102
Verandah Deck 7 midship across from 7267
Dwayne