The door ajar light on the dash stays on all the time. Even after making certain that all doors are properly closed and tailgate is secure.The dome light stays on also.

The switch or wiring at one of the doors may be bad allowing light to stay on .

check all door switchs the most comun is the driver side maybe is fail or disconected and fuses too,

Check for corroded wire connector kick pad driver side clean wire try this or change the connector

i must say that i feel your pain with this,, my door ajar light does the same thing but my dome light doesn't come on when the doors are open, i have to flick the switch on the dome light to make it come on. In my opion and i stress my opion it saounds like a issue with the body controller, im not sure what ford calls it but it controls all the body functions, i dont think the low coolant light is connected to the body contoller but i dont know this for sure, i think that light is controlled bny the engine ecm. some times a dealer can connect to the body controller and see if there are any fault codes for circuits that it controls and or monitors, also they can reflash or update the configeration and or calibration level cause ford release's fixes for issues all the time, same theory works for any control module on a vehicle. i dont think any of the auto parts stores have the right program to connect to bofy controlers but i could be wrong, worth looking into. i only put up with mine cause i use it as a beater, if you are worried about your dome light staying on and drianing your battery then you could pull out or disconnect the dome lights for the time being, its not a fix but just a band-aid.

The car might be "thinking" that one of the doors is ajar,
Check the spring switches in all the door frames to make sure that they are hooked properly, and doing their job. One way to do it is ( if the dome happens to be off) to close and open each door several times, to see if the bad switch stays open and turns the light on, even with the door closed

To temporarily solve the light problem, I taped a washer to the inside of the door where the door hits the plunger on the light switch. A permanent solution would be to drill a hole in the door at this spot and put a self taping screw in that will trigger the switch sooner. I figured out the reason the dome light stayed on is due to the courtesy light timer and the lights would have eventually gone off.

The door sensors will need to be replaced. A screws can be removed and then a wire can be taken of and you then can replace the switch.

spray the door latch with WD - 40 on both doors then open and close multiple times. Sometimes they get stuck ond this help if they free up once a month should keep it away

Have lost power to the passenger side door. Without power system doesnt recognize door closed because of no power to door ajar switch which keeps dome light on. Security light stays on because without power to rest of systems supplied by fuse it cannot tell if doors and windows closed to engage security features. Most common is for power wire to break somewhere near where it leaves to main body and enters the door.

If I am not mistaken, the plunger is located in the door with the latch. Most of the time you can just pull the door panel off and adjust the switch. After opening and closing the door, they get knocked out of adjustment. I sincerely hope that this helped you. Take care and best wishes.
Bryon