A half diminished chord included a b5 and b7. If you're playing in drop D you're probably playing some variant of rock so I doubt you'll run into them too often

You've got the right idea though. Just remember the fifth is up (in pitch) by one fret if it's augmented and down by one fret if it's diminished. The best way to get used to the "odd" dropped D is by playing scales starting with a note on it. Eventually it will become second nature and you won't have to think about it.
p.s. in doing arpeggios with drop D I find it more comfortable to do the 3rd also on the D string to avoid strange stretches.