You didn't say what year you Explorer is. Mine's a 2001 and here's what I've learned...
Each door and the hatch have a small switch which is attached to the latching mech. The switch is a very simple push-button device which is supposed to be depressed when the door in question is completely closed. These switches can get stuck and they can easily break or just wear out. My hatch sometimes leaves my dome light on and also sets the door ajar light, even though the hatch is closed. Opening the hatch and reclosing it fixes my problem, but I've also had a broken switch in the driver's door, which caused the dome light to not come on, the "door ajar" light to not come on, the "key left in ignition chime" and the "lights left on chime" not to work. In your case it sounds like you have a switch which is stuck/sticking. This can sometimes be fixed with some WD40 once you figure out which one it is. If you need to replace a switch they not very expensive (about $8 from Rock Auto) and changing them is easy requiring only a Phillips screw driver once the door panel is removed. I think your rear wiper issue (another common problem) is unrelated. I'd start with checking the driver's door switch as it tends to receive the most wear.
