When I turn the key off, it is like disconnecting the battery.

Sounds like you have a power leakdown over night, after all night the engine and battery is cold causing the issue.
does the terminal spark when you connect it?? If so something is dragging power..
Check none of the accessories are drawing power when key is off e.g radio, cig lighter adaptors, interior boot lights ect....

You replaced the cig lighter fuse ?
That should have fixed the radio problem if the lighter circuit has power now. Keep in mind when you kill the radio memory it resets to AM and not on any pre-set station. Just like when you disconnect the battery cable.
The radio gets memory power from the cig lighter circuit and regular power from the radio circuit thru the key switch.

your control module is faulty..

you need to check for a bad fuse in the fuse box in the engine compartment.

What did you plug in? I had a similar issue when I plugged in a high amp tire pump. It kept blowing fuses because the pump itself had a short. Be wary of using stuff like that in the cig lighter. I switched to a hand held pump. How was the computer found to be defective? I wonder if the cig issue shorted the computer?

sometimes that works, sometimes not. check the lighter if theres power with the key off, or if your cell phone charges with the key off. If not, its not going to keep the memory. I would take 2 jumper wires and connect the new battery to the cables, so that the car never loses power. or take jumper cables from another car and connect to the battery terminals.

The headlight switch could be damaged but to disconnect the radio first as some radios that did'nt come with the car may be connected wrong and power is going from the power wire at the radio to the dash light circuirt feeding thouth it to the regular lights. pulling the cig lighter fuse solves the problem points you to this answer.

Check the connections to the battery terminal. Then check the fuses. There may be a "master" fuse.

The memory wire of the radio,(the power for the clock and channel settings) is usually connected to the same circuit as the cigarette lighter. (there is a seperate fuse for radio) Check your cigarette lighter socket for damage or dirt, etc. could be causing the memory fuse to blow. Also check your cig lighter accesories (like cell phone chargers,etc.)

P.S. The backlight is also a seperate circuit (connected to interior light/parking light circuit)

CHECK THIS....IT'S YOUR IGNITION STARTER SWITCH....STEERING COLUMN