Taurus hesitates when engine hot or weather hot?

My 2000 Taurus SE Vulcan is acting a little weird, depending on the engine temperature. When the weather and engine are hot, the car hesitates when accelerating or cruising (40km/h-60km/h) under normal driving conditions. Acceleration is not smooth, and heck the speedo needle even turns in a jerky manner.

Now the weather is cooler (6C-12C) when I leave for work, the engine runs smoothly when cold, with no hesitating. The speedo needle seems to turn much more smoothly than when the engine is hot.

It's only when the engine begins to warm up that it stops running smoothly.

We put some fuel cleaner through a week or so ago, and gave the engine a bit of hard acceleration at 80-100km/h, and things seemed to improve a little bit. But now it's running very un-smoothly (is that a word?) again.

A friend thought:

"Sounds to me like the ECU thinks the engine is cold all the time"

But I don't know anything about engines, so I can't tell what this might be. So any suggestions greatly appreciated.

Thanks