The reverse lights get power from fuse no.7 (15 amp) in the under dash fuse panel on the gy/rd wire. They get ground from the GEM module on the dg/og wire.
Do you have power to them?
Yes it could be that or it could be the REM module is not grounding them.Have you checked to see if you have power on the GY/RD wire at the reverse lamp sockets?
If so try grounding the DG/OG wire and see if the bulbs light then.
Here is the wiring schematic for the reverse lamps
The REM is located behind the right rear quarter trim panel here is a pic of the location
I have circled the connector locations for you.Now so you know this is a programmable module or computer.It is programmed to your vehicle. About all you can do at this point is check the wiring from the REM to the bulb sockets. The REM gets the signal that vehicle is in reverse from the PCM which gets this from the transmission range sensor that is attached to the transmission linkage.The PCM receives a voltage input from the transmission sensor then it sends this signal out of the network to the REM module which will then ground the bulbs.
So to make a long story short you would need a scan tool that can communicate with the modules on your vehicle and can look at data to see if these signals are being received and sent.
Here is a pic of the connector you will need to access on the REM to check the wiring from the REM to the bulbs. All you really need to do to check them is to ground them like you did at the sockets and if the bulbs light they are good.
Here are the instructions.
6. Disconnect the shift cable
7.Remove the manual control lever
8.Remove the digital TR sensor
3.Using the special tool, align the digital TR sensor
4.Install the manual control lever and nut
5.Connect the shift cable.
6.Verify that the shift cable is properly adjusted
7.Connect the digital TR range sensor electrical connector.