Mercs are the only cars I've owned that tend to run cold when the thermostat fails. A 380SL and a 190E both suffered this way and a new 'stat fixed both. Cheapest first step possible.
Next, the fan. If I remember correctly, the 380 has an engine driven fan. These can fail by losing drive in the clutch, or locking up. If it locks up, it will over cool.