If you are putting the test leads from one terminal to another on the switch with wires connected, you may be testing hot lead on one side of the switch to ground on the other side of the switch.
I suggest you disconnect the battery (always a good idea), disconnect the wires from the terminals on the switch, and then test the switch with the key off.
When the key is off, you should not have continuity from the battery terminal to the ignition terminal on the switch. With the key on, you should have continuity from the battery terminal on the switch to all other terminals, depending on the key position.
If you have no continuity on the switch with the key off and the wires removed, something else is wrong. The ignition switch may be bad, in that, once you turn it on you can't turn it off, somewhere you have jumpered around the switch in some way, or some wire(s) are bare of insulation and touching.
With the key off and removed from the switch, do you have power to the gauges? Does the fuel gauge deflect? Does the ammeter deflect? Do you drain the battery after a while?
