I'm on a Mac OS X 10.5.8 and I'm having the same problem.
If I just drag calibre 0.6.12 from the mounted dmg to my Applications folder and have it replace the older version of calibre, I can't get calibre 0.6.12 to work at all.
If I uninstall and delete the calibre related files, and then install 6.12, calibre will open with the wizard and everything goes smoothly. But I cannot get it to open a second time. I have to uninstall calibre and reinstall it to use it again.
I'd appreciate any help to fix this. But for the time being I reinstalled 6.11 so I can use cailbre.
ETA: Nevermind! I got it to work, I just had to delete this: net.kovidgoyal.calibre.plist then do a fresh install of 6.12. Everything seems to working fine now.