My 2004 malibu has a mind of its own....

The servo motor in the door lock may have failed, broke, or disconnected. An Authorized Chevy Dealer should be able to check it.

The problem is possibly a ground failure into the computer

If you don't have a key fob you're out of luck. It is needed to turn the alarm off completely. You can shut it off by using the door key to unlock then re-lock the door, but the alarm is still armed and will go off if the passenger door is opened after the drivers door is locked

The locks on your vehicle are electronically controlled by the BCM - body control module . The driver an passenger door switche's are input's to the BCM . Do you know anything about automotive electronic's ? anti theft alarm remote, could this be part of the problem , No it's just a key less entry remote , nothing to do with the anti-theft .
Door Lock System Components
The Power Door Lock System consists of the following components:
• Body control module (BCM)
• Driver door lock switch
• Front passenger door lock switch
• Door lock actuators in each of the doors
• IBCM (PWR) 15-amp fuse

Door Lock System Controls
The Power Door Lock System is controlled by the body control module (BCM) and can be activated by any one of the following:
• A door lock switch Lock or Unlock actuation
• A keyless entry transmitter actuation
• An automatic door lock actuation
• An automatic door unlock actuation
• A last door closed locking (LDCL) actuation

Driver and Passenger Door Unlock Operation w/o Keyless Entry
When the body control module (BCM) detects a transition of either of the door lock switches to the Unlock position, it switches the door lock actuator unlock control circuits to voltage. Since the door lock actuator lock control circuit is at ground potential within the BCM, the driver passenger door lock actuators transition to the Unlock position.

Your best bet , take your vehicle to a qualified repair shop ! Need special tool to diagnose .

Power Door Locks Inoperative

Install a scan tool.
Turn ON the ignition, with the engine OFF.
With a scan tool, observe the door lock switch parameter in the Doors data list.
Does the scan tool display Idle?
YES - Go to Step 4
NO - Go to Step 8
4
Activate the driver door lock switch to the LOCK and UNLOCK positions.
Does the scan tool display Idle?
YES - Go to Step 9
NO - Go to Step 5
5
Activate the front passenger door lock switch to the LOCK and UNLOCK positions.
Does the scan tool display Idle?
YES - Go to Step 9
NO - Go to Step 6
6
With a scan tool, observe the driver door lock actuator command and the passenger door lock actuator command parameters in the Doors data list.
Activate a door lock switch to the LOCK and UNLOCK positions.

Open the drivers side door and see if you can peel back the covering on the wires going into the door. Sometimes from opening and closing the door so much 1 or more of the wires break or rub bear and contact each other or something else. Check if all the wires are good.

Go to pass/side door and use key to lock and unlock door and this will shut down alarm

The alarm is self setting when you lock the doors it sets it. buy a fob from the dealer That is what I had to do.

have you checked or replaced the starter relay?

I think thats normal, if you turn off the engine, you should be able to open /close the door locks.

had very similar problems with a Hyundai coupe , turned out to be the central locking control mechanism , this is usually the drivers door lock unit inside drivers door , this controls the locking system and so also interior lights as if having used remote etc,hope this helps