The right use of both is
Both X and Y where both X and Y should be parallel to each other meaning both of them should be either verb or noun.
With that in mind:
Quote:
A. Doctors both
espouse the benefits of the influenza vaccination and
advocate its administration
-- both espouse and advocate are verbs and parallel, also its unambiguously refers to the influenza vaccination
Quote:
B. Doctors both espouse the benefits of the influenza vaccination and
are advocating its administration
---- changes the meaning of the sentence, saying doctors do one thing and are doing another thing right now which isn't the case
Quote:
C. Doctors, espousing the benefits of the influenza vaccination, advocate
their administration
-- their refers to Doctors, hmm so doctors are administrating themselves.
Quote:
D.
Both espousing the benefits of the influenza vaccination, the doctors advocate its administration
-- Where is AND, I can't seem to find it
Quote:
E. Espousing the benefits of the influenza vaccination,
its administration is advocated by doctor
-- Wrong modification, Espousing should modify doctor not the influenza vaccination administration