Most adapters these days are switchmode types, which means they are regulated, ie their output voltage doesn't vary with the load.
Since you say that your drive uses a 12VDC adapter, then the most likely culprit is a shorted TVS diode on the +12V rail. You can simply remove this component with flush cutters.
Assuming there is no other damage, the drive will work perfectly well without the diode, but it will no longer have overvoltage protection on the affected supply. So be absolutely certain your power supply is good.
If, for continued protection, you wish to replace the diode, then a 5V TVS diode can be substituted with an SMAJ5.0A, and a 12V diode with an SMBJ12A. Both can be ordered from Farnell, Mouser, Digikey.
These URLs should help you identify the components:
http://hddscan.com/doc/HDD_from_inside.html
http://www.users.on.net/~fzabkar/HDD/HDD_ICs.txt
http://www.users.on.net/~fzabkar/HDD/TVS_diodes.txt
If you need more help, then upload a detailed photo of the component side of the PCB.