My dome light bulb died do I decided to try one of those LED ones in the store since it was supposed to be brighter. Maybe without the lens it would have been.

With the light is on, it works properly. When the door is closed and the switch is off, the 2 middle bulbs came on, but at very low intensity.

Is this normal, or do I have a wiring problem somewhere? I'm guessing the low draw isn't enough to light up an incandescent bulb at all but just enough for the LED to light a little.

 

That's a possibility. Electrical interference can also cause an LED to come on very dimly. I have this problem with the high-beam indicator LED in my custom dash. If the car is running, the electrical "noise" from the ignition system will keep the LED dimly lit....not enough to see during the day, but just barely noticable at night. Oddly enough, the turn signal LEDs which are right next to it don't do this.