2008 F350 noise with engine off
 

My 2008 F350 is making a whining noise with the engine off. It drained the batteries over the weekend. Had to jump it. Took it to dealer yesterday. They replaced the batteries and couldn't reconstruct the noise. I drove it home, turned it off and there it is. High pitched, whine from center of engine. Disconnect batteries and it stops. Restart engine, turn it off again and whine is back.
After disconnecting batteries to stop whine, reconnecting them does not restart whine. Only restarting engine then turning it off causes it.
One forum suggests stuck fuel pump relay associated with this symptom.
Bummed that I'm headed back to the dealer tomorrow as it is very inconvenient for me and takes a couple hours of my time just to transport it.
Any suggestions?
Using University Ford in Durham, NC
Thanks, B