I'll make it short and sweet: If I have my guitar tuned down one step to D and am using a melodic minor scale in E (12th fret), would the shape of the melodic minor alter at all considering I'm tuned down one step? Alternatively, if I am using the melodic minor shape in E, would I simply move it up to the 14th fret? I hope this isn't too confusing.
If you're tuned down to D and starting a melodic minor scale on the 12th fret, lowest string, that note is D and hence you're playing a D melodic minor scale.

It's still the same shape as you learned before.