Yeah, I just used the ground wire on the harness. That will be more than enough. It is supposedly better to ground it yourself, but it won't matter unless you have a million components grounded in the same location (can't remember why, something about a ground loop, but it was a good source anyway). If you want to ground it yourself, you should find a metal spot that has a screw in it. Get a ring terminal from Radio Shack or somewhere and solder (you can crimp too, but I prefer solder) that to the ground lead from your head unit. Then sand the ground location a little bit to make sure you have a good connection, wipe it off, and screw it in.
That should take care of the ground, but it still doesn't explain why your unit shuts off and beeps. That means it's still getting power, but thinks you took your key out. I suspect that you may have messed up the connections when doing the wiring harness. Check the Accessory (ACC) or Ignition (IGN?) wires because they are the wires that tell your unit that the car is on. When it is properly wired, as you know, the unit will beep when you turn key to OFF while face is still on. So, that's why I am thinking that one of those signal wires is faulty.
I'm afraid that the $30 may not have encouraged them to check wiring harness. Did they?
If you have a soldering iron, try that. It is the best way to connect electronics. Very strong, and very good electrical connection. If not, just redo the crimps.
Good luck. Let us know how it goes.
Also, if you have a multimeter, check continuity between head-unit side of harness, and vehicle side. That will help you diagnose your problem very very quickly. If you don't have one, find a friend, or get one at Radio Shack. You can get one pretty cheap I believe (~$20).