Absolutely, you should vary your wrist position. The wrist is one of the most important joints well, actually a whole set of joints) in your playing apparatus, in addition to the finger joints, elbow, shoulder, torso

you get the idea. None of these joints should be held in a fixed position throughout one's playing. They all need to be involved and should constantly be adjusted according to the sound you want. of course, certain, short passages will require one or more joints to be held constant, but this is not to be applied to any passage.
Furthermore, the wrist position you described (i.e. back of the palm in one plane with the forearm) is not high at all. A high wrist position is with the wrist a lot higher than that. The "natural" position, which is already higher than what you described, is where elbow, forearm, hand and fingers form an arch.
One example for a typical wrist action is the general way for playing a phrase: start with a low wrist, move the wrist up during the phrase, and lift off the hand at the end of the phrase with the wrist at its maximum height.