Welcome to the board! I hate electrical problems but have gotten pretty good with a multi-meter. Do you have a multi meter and know how to use it?
A little clarification on what you posted will help.
I was wrong, first of all when the key is turned to the OFF position the car alternator remains on.
Do you mean the car keeps running or you believe current continues to pass through the alternator?
I have to turn the key between off and start to get the car to be fully off but I can't remove key in this position obviously. Sometimes the car turns over but most of the time it doesn't... seems like it drains the battery immediately.
When you say 'fully off' what do you mean? Are you saying this is what you do to get the car to stop running? Are you moving the key from 'off' to 'start'? The car not running over symptom happens when you are having trouble turning it off or all of the time?
Before you pull the ignition switch let's confirm you are wired correctly at the starter solenoid. It is easy to swap the two small wires and it will cause strange results. There are 4 posts total, two large and two small. The small one near the battery gets power from the ignition switch when you are in the 'start' position. It energizes the solenoid and completes the circuit to the large post that is furthest away from the battery. This goes to your starter. At the same time it also completes the circuit the to the other small post. This goes to your coil and provides non-restricted current to the coil during cranking. Once the engine starts the circuit to the starter and the coil is opened. The coil continues to get power via a resistor wire.
With a multi-meter confirm voltage at the battery. I should be close to 12v when the car is not running.
Determine which of the two small wires gets approximately the same voltage(about 12v) when the ignition switch is in the 'start' position and only in the start position.
The other wire may indicate voltage while the car is in the 'run' position but it should be less than 12v if a previous owner did not remove the resistor wire.
The wire that only has voltage while in the 'start' (crank) position goes to the small terminal on the solenoid that is closest to the battery. If you look closely it might have a 's' indicator. The other wire goes to the other side, if you look close there might be an 'i' indicator on the solenoid.
Here is a picture I found of the solenoid.
Once you have confirmed this see if your problems continue. If they do, then try removing the wire from the 'I' terminal and see if that fixes your problem (for trouble shooting purposes)
Does the engine look modified? Can you tell if there is an new/aftermarket coil? Does it look like someone has rewired the engine compartment?