Spaw, what I really needed from you was the question, "Nurray, what's Gaelic tuning?".
Lane, when you play the open (unfretted) strings in Open C you certainly don't get a D chord. You get a C Major chord. The tuning you are using is a variation of Open C tuning, known in my neck of the woods as C modal. To get a true Open C you leave the first string tuned to E.
There is nothing wrong with the tuning you are using, many, in fact most Celtic guitar players use CGCGCD rather than CGCGCE. I don't personally, but Tony McManus does, so go figure. For your purposes (singing in D) I would suggest that you capo up on the second fret and play the C chords.
When I have more time I will send you a list of diagrams of the chords you need (using your CGCGCD tuning). Meantime, off to work.