As it happens the last tune example I've added to my tutorial is "Summertime". It's in A minor/dorian hexatonic, but if I understand what Gershwin's up to in the accompaniment, he mostly treats it as Dorian, and there are frequent Am6 chords to keep that F# in your mind, even though it isn't in the tune. Lox will have a better feel for what his intentions were.

More straightforwardly an A dorian tune would have IV(Dmaj) instead of the iv(Dmin) you'd get in minor, and ii(Bmin) instead of iio(Bdim). A progression like Bmin > Dmaj > Gmaj > Amin might be reasonable in dorian and wouldn't be in minor.