Sounds like the thermal is cycling. Check it and clean out debris. (overheating)
Sounds like a bladder tank issue. The bladder gets potential energy when the pump fills the tank. When you open a faucet for water, the potential energy stored in the bladder will push the water out of the tank without the need of the pump coming on until the tank is almost empty. You should be able to rock the tank when the pump first comes on to see if it is light weight, as it would be when almost empty. If not then there's your problem. When the pump ends its cycle is when the tank will be at its heaviest. Again: The tank should be full when the pump is not running and almost empty just before the pump comes on. The water pressure would behave as you describe if the bladder tank is not functioning properly.
