This is not you basic dimmer circuit......
In the old cars, you would feed a dimmer module and it's output would send power directly to each bulb on the circuit....
Not the case with a 2003 Limited.....we are talking modules here folks.......and these modules communicate via serial data.....
For example, the dimmer switch sends a variable voltage to the IPM.....based on that, the IPM(instrument panel module), sends out a signal via Class II serial data to the Dash Integration Module, which then sends out a variable voltage to the Steering wheel, HVAC, radio lights, etc.......
There is a 10A DIMR fuse that powers these lights....that's about all you can easily check(rear fuse block).....
You need a Tech II scan tool to read "B" and "U" codes to see if they exist.....and if you are good at diagnosing with electrical schematics, first place to start is a schematic of the Dash integration module.....
Possible problems are, no power to DIM, no communication between IPM and DIM, bad DIMR fuse, an open circuit, bad DIM, etc. Like I said, not an easy system to diagnose.....
If determined to fix it on your own, first thing to do is get a schematic of the DIM in the dimming circuit, test light, and DVM......disconnect the harness, and check for voltage in and ground, and also that you are getting serial data into it....
You can also jumper the voltage, in to the DIM, to the output circuit of the DIM(via the disconnected harness) to see if the lights go full bright, just to test that circuit....