Welcome to the forum
I have to ask so don't get mad..............is the dash light dimmer wheel turned OFF for dome light, it has a separate switch inside to activate dome light when wheel is turned all the way "up", you can feel a click on and then off when turning the wheel
I assume the "Door Ajar" light is OFF and only dome light is on, if thats the case then its not a GEM module issue
Above the gas pedal is Relay Box 1, it has a black cover over it, it faces towards parking brake pedal
Inside that box is the Interior light relay
The GEM Module activates this relay when a Door Switch is grounded, it also turns on "Door Ajar" light, at the same time
So no Door Ajar light means not a GEM issue
If there is a door ajar light then one of the door switch wires is shorted to ground somewhere or GEM is bad
Interior light relay is double pole so when not activated the dimmer wheel can send 12volts thru it to turn on dome light, this will also turn on the Cargo light on top back of cab if you have one(part of 3rd brake light)
Relay box 1 has a few relays inside and there is no good MAP for it because lay out changed yearly AND changed with Options added
After you take the cover off the relay 1 box see if it looks like this:
http://ww2.justanswer.com/uploads/sasso1179/2010-07-12_195044_relays_one.gif
If you pull battery saver relay OR interior light relay out dome light should go off
Wiring diagram below