Hello, I hope the following assists you.
As the doors are unlocking again straight away but you say all the motors are operating including the side and rear doors then this suggests that a door ajar circuit has failed so the central locking control module detects a door as open even though it isn't.
The most common doors to give problems are the side and rear doors, the side door tends to have alignment concerns as the middle roller collapses which means the contact pins don't align correctly. The driver's rear door suffers with wiring problems where the wire in the door bets damaged by the constant opening and closing of the door, if the ajar circuit breaks then this will show the rear door as open.
Unfortunately without Ford IDS you can't easily confirm which door is been detected as open, it can be more difficult as well as the rear door wiring becomes all black where it goes to the door as a security measure.
All the ajar switches are integral to the latches apart from the sliding door so you can disconnect the connectors and check the door latch ajar circuit operation, pin 8&6 on the rear door, pins 8&6 on both front doors. The sliding door switch should have only 2 wires to it.
Try putting the side lights on with the ignition off and leave the driver's door open, if the buzzer doesn't sound this would suggest the driver's door is at fault. The buzzer doesn't work from the other doors.
I hope this helps but if you have more questions let me know and I'll reply as soon as I'm able to.