I had the same problem 2 years ago on my 2003 Impala LS 3.8L
The fans would run all the time. It finally got to the point the bearing in the fans started to whine from over use, so I replaced the Engine Coolant Temp Sensor and the fan assembly, this seemed to work for a month and then they ran all the time.
After a year, I finally had a squeel from the fans again, I got a fan assembly again, replaced the unit and found a blown fuse for Fan1 under the hood near the pass. side firewall. I need to test the relays as well, but so far the fans don't run at all - which is scarry........The only difference with this new fan asbly is it is a Dorman, the old one was a Siemens.
Let me know how you made out though....
Also, I am having the instrument cluster problems most other Impala owners are having. At first it was only the speedometer and gas gauges, but today it is the gas, speedometer, engine temp, RPM and voltage, the oil pressure gauge seemed fine....I assume this is means the entire cluster needs to be replaced?