If the meaning of the word
Verity is part of why you like the name, both True and Truth have been used as given names on rare occasions and the name
Alethea also goes back to a word meaning "truth."
I do agree with the others who say that "V.D." is a now a very out-of-date term and children your daughter's age are unlikely to know it, so I think
Verity would actually be fine.