You really should be "bothering with the wiring" at this point. CCRMs and Temp sensors are pretty simple devices; they don't fail that easily. I personally replaced a CCRM on mine when dealing with an A/C clutch that wouldn't come on, and it turned out to be the clutch connector (a $14 part). I've also had issues with my fan connector that caused the high speed fan to fail to start.
Check the fan plug first; in my case one of the wires had started to back out of the connector (the rubber seal on the back was out just enough to break the circuit). But, it's also a high amperage circuit, so damaged/burned/etc fan plugs aren't that uncommon.
If the connector is good, get a multimeter and start testing the wiring. Even a simple tone test would be helpful. Some information that might help:
Wires at the Fan plug:
Black - Ground
Red/Orange - Low Speed +12V
Orange/Light Blue - High Speed +12V
Wires at the CCRM:
Red/Orange (2 wires, should be pins 1 and 2) - Power out to Low Speed Fan
Orange/Light Blue (also 2 wires, should be pins 6 and 7) - Power out to High Speed Fan
Yellow/Black (again, 2 wires, pins 3 and 4) - Power in from Junction Box (50A fuse under hood) for fans
Check for continuity from the CCRM to the Fan (on the Red/Orange and Orange/Light Blue wires at each end).
Check for power at the CCRM (on the Yellow/Black wires).
With the engine hot and running, check for power at the fan plug (low or high should be on, but not both).
This definitely can be a little time consuming, and annoying, but it's difficult to repair this sort of issue without proper diagnosis, so the time is worth it.