Sorry Brian ,
the version i sing starts off clearly in c major, the opening notes are e f, then we have the first bar line then g e edc, that is quite clearly a c major chord , bar 2 then could be harmonised two ways by g or by d minor.
the tune i sing does start in a major key and then chnge. bar one is harmonised with notes of a c major chord
the change to a minor chord can be in bar 2 or bar 3
personally i prefer he first d in bar 2 to be harmonised with a single g note and the next d note to be harmonised with a single low aor possibly g , so i am suggesting chords rather than playing full chords.
opening notes of the version i sing are not in the dorian mode.