If you've enabled crossfire in the CCC and it is reporting crossfire enabled, you can cross the following problems off your list:
(1) bad crossfire connector
(2) bad power connector
Also, you can cross off old CAPs (since even without CAPs you can run 3dMark11).
In my opinion, this is a driver problem. Now, either AMD has its head up its *ss on the 7800 series, and that's why we haven't seen a single review of 7850 crossfire, or, less likely but hopefully, it's just a bad driver installation.
I'd remove one card with the PSU power cable disconnected, restart, delete all drivers and run driver cleaner, and shut down. Install second card, connect all power/crossfire cables, and boot. You'll then have to install 12.4 (I don't think 12.3 supports the 7850, does it?). Restart. Install 12.4 AGAIN, just to make sure it catches the second card (that this helps might be an InterWebz myth, but try it anyway).
Report back.