2002 Ford Taurus Heater Problems

[Xray]

I have a '02 Ford Taurus with the Vulcan engine. It has 98,000 miles. I starting having a heat problem recently.
The heater when idling blows luke warm. I noticed as I accelerate when driving it blows hotter. Stopped at a red light at idle it blows cooler (not cold just luke warm or barely warm).
When accelerating, I noticed that the temperature gauge would drop down to the cool side. Not all the way down but would drop enough to notice. As soon as I would stop accelerating the gauge would go back up.
I pinched off the bypass hose by the heater core, I noticed that the radiator hose going into the thermostat was cold. Heat was blowing a little better, temperature gauge was normal. As soon as the clamp was taken off the bypass hose, the radiator hose to the thermostat got warm. The temperature gauge dropped down to the cooler side for a few minutes then went back to normal.

I have been researching for a few days and got allot of things that could be wrong. Any ideas from the symptoms I gave? Thanks for any help.
Long time reader first time poster.

[mmsstar]

The Vulcan engine is known to have water pump issues.

[Magus732]

The Vulcan engine is known to have water pump issues.

"Known" by whom? When the water pump on my 2000 FFV Taurus died back in late Febuary during that cold snap in central Ohio, mine was already at 140K+ miles. Seems like a long time for a "failure prone" part to work before something goes wrong with it.