Hot and cold shower

Your pump is reaching shut off pressure and turns off then pressure drops and pump turns back on. Cold water system has less restriction then hot so when pump turns back on cold flows first then hot catches up.

Either the pump pressure switch is set too low or the flow restrictor in shower is causing the pump cycling.
Remove shower hose from faucet and remove the restrictor in end of hose (may be in handle of shower wand). Might even have to drill it out to a larger size.

If it continues to short cycle..then you can increase the pressure set point.
On the end of pump center of head is a screw. Turn it 1/2 turn clockwise then check pump operation. Don't turn adjustment screw more than 1 1/2 turns total.

Folks have reported good showering with an Oxygenics Shower Head

Just for giggles...check the water heater valving.
Make sure the bypass valve is closed. Bypass valve connects cold and hot lines right at backside of water heater.

Are you running the pump while on city water

First, is this operating from the tank or on "City water" with the pump On? If on city water you don't need the pump ON. If operating from the tank I would check the outside shower faucet and make sure both hot and cold faucets are OFF. If they are left on and the shower head is turned off at the head itself, hot and cold water can mix through the valves and cause cold water at your inside shower. Let us know which way you are operating, from tank or city.

Turn off the water pump when using city water.

Sorry I did not clarify, I am not on city water. Using what is in my holding tanks. And the outside shower is off. It only seems to do this in the bathroom shower.

So you get steady hot at the sinks?

bypass valve is closed. I do get steady hot at the sinks. And I put in an Oxygenics Shower head, with the same hot and cold results. gah! frustrating!

I have the same problem. I also have the Oxygenics head. I felt if head flowed more water it wouldnt happen. Mine is not that bad but wish it wouldnt happen.

Oxygenics don't flow more water they just mix cold/hot better plus add air to mix.

With the shower head removed & just hose attached to faucet does it react the same way.....pump cycling.pulsing hot/cold??

I also have a Oxygenics shower head and this is how I use it. Turn on the HOT faucet full blast first, then after the hot water flows, turn on the cold faucet just as far as needed to get the right temperature, get in and shower. Works for me.