Shower water temperature fluctuations

While dry camping, our shower water temperature fluctuates with the water pump cycles. When the pump is running, the temperature stays as you set it but when the pump pauses it only pumps cold water. When the pump comes back on, the water temp returns to what you have set. We added an accumulator tank which helped a little but did not eliminate the problem. I've thought about replacing the cartridge in the control but can't see how that would cause this problem. Has anyone had this problem and, if so, how did you solve it? We have a '99 Safari.