A/C vent temperature fluctuating?

This happens when the rheostat inside the blend door actuator goes bad and is not allowing the controller to be stable. Here is the location of the part so you can check it out. There is no simple test for this, I would try a new part and see what happens. Check out the diagrams (below). Please let us know what you find.

I see. So, what happens if i unplug the left one while the air coming out cold? Will it just stay there?

Yep, it will stay on the cold setting. Not a bad way to go to get the cold air in this heat.

Does the panel around the carpet need to be removed? I cannot tell from the pictures.

I unplugged it and went all the way to very hot air.

Hmm, it must need voltage to stay in one position, it was worth a try.