I just played through the Shivering Isles for the first time and found that I'm still one achievement short of 100%. Is it possible to get both the Duke of Mania and Dementia achievement with the same play through?

I don't think so, I just made a save before the part that pertained to this part.

nope

Save before completing the quest to become Duke for one side, do it, get achievement, load, do it again for the opposite side. That isn't what I did but I see no reason why it would not work.

I saved before entering one gate, entered it, got the achievement, reloaded the save, then entered the other gate. Both achievements unlocked in one playthrough.

+1 Achievement
-1 Honor