Nick, I personally don't play in a droning style, so those who have already responded are more help than I can be. And they've also given you solid advice. In the key of D, DAd or DAA would be the most common tunings. In the key of G, DGd would work. And Ken's advice on bagpipe tuning might work in either case.
Another possibility, though one shunned by traditionalists, would be to use a capo. If you are tuned DAA or DAd, you can obviously play in D. With a capo a the 3rd fret you can play in G, and with a capo at the 4th fret you can play in A. The other common key would probably be C, and you could tune down a note to CGG or CGc for that.
The main advantage of using a capo (the fact that your fingering doesn't change as you change keys) is less important to drone players than to us heretical chord players, but you might consider the option if it doesn't offend your traditionalist sensibilities.

.
It's silly (of course), but it has really opened some doors and I offer my sincere thanks to them for making me see it through.
I never enjoy it when we wind up playing in a session with a guitar player like this.