I'll add to the water temp speculation.
If you have air in the top of the HW heater it can compress when the pump comes on, causing differences in pressure/volume on the hot side at the shower head. The cold doesn't have air and therefore doesn't compress (hydraulics) causing temperature differences while the water is on.
You wouldn't notice this problem if on city water as it has fairly constant pressure.
At the pressure relief valve, check it and let any air come out. (open it just a small amount)
I usually do this when I put water in the tank but have let it out with the heater on.
Be care full. (Don't burn yourself)
"Just speculation"
