Man=אדם, Person=איש/אישה. "Person" refers to personality (אישיות) and has no gender: ex: "She is a sensitive person" "Man" refers to "male" (גבר) and "human" (בן-אדם/בת-אדם) so it applies the gender inside.
Both are correct, your answer is literally more accurate (Dad=male)
I wouldn't say that was more correct - it just means something different.
"My father was always a sensitive person" = My father was a sensitive person, with no information given about the present. Most likely explanation - my father died. (Possible but less likely explanation - my father is still alive but is no longer a sensitive person. I would expect that to be expressed as something like "My father always used to be a sensitive person" though.)
"My father has always been a sensitive man" = My father was a sensitive man in the past and still is one. Inference: He is still alive, and he is still a sensitive person to this day.
TL;DR: "My father has always been a sensitive person" is an absolutely correct English sentence, but it isn't a better version of the "always was/was always version, it's a different sentence.