The battery is still good and the motor will keep running when you disconnect a terminal. Can my alternator be bad? If not I'm also thinking a dead short? What is the best way to find out?
Sounds to me like you are not making connection to the battery (maybe this is what you mean by "dead short"?). I would look at your contact points ... both sides of the battery, ground, and starter to ensure you are making good contact. If you have side posts (assuming you do), these have a tendency of not making complete contact even when tightened. Make sure there isn't any rust or corrosion at the ground point causing interference. If you can disconnect a terminal, the alternator is still putting out juice. You can also put a multi-meter on it to test its output.
Also, when you say the battery is dead, do you mean you turn the ignition off, then go to turn it back on and ... nothing? Or does your dome light/radio turn on? Do accessories die when you turn the key or dead from the start? If you have dome lights which die when you turn the key, it's the connection at the battery as described above. If you turn the key, still have dome lights and nothing happens, it could be your starter solenoid or your ignition switch.