Unless you have one of the new computerized pumps, surging is not all that unusual at some flow rates. The pump comes on at full power when the low pressure limit is reached and then shuts off when the high limit is reached. If the pump output rate exceeds the demand, the pump switches on & off and the pressure "surges" between the high and low limits. But if the pump stays on and the observed water pressure at the showerhead changes (surges), there is a problem somewhere.
The fact that you get a temperature change makes me suspect that you have crossover between hot and cold lines somewhere. One common cause of this is a mixing faucet such as an outside shower. If the water valves for an outside shower are open and the water merely turned off at the shower head, cold and hot will mix through the valves. Always turn off the outside shower at the faucet when finished using it. As others have mentioned, a partially open water heater bypass valve can also cause mixing of hot and cold.
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL