Some synonyms may be used in place of "carry". To carry on has the same sense as to continue, to proceed or to keep on going, for instance.
We could rewrite the sentences in the following way:
Carter Druse keeps on going without even the smallest thought... Almost emotionless, Carter simply proceeded with his watch.
or
Carter Druse goes on without even the smallest thought... Almost emotionless, Carter simply continued with his watch.