Help! Water pump pressurizing when using hot water tank.

That new pump is a bypass type pump- it has an internal bypass valve that recirculates water at low output demand. The idea is that instead of rapid cycling when you open a 2 gpm faucet with a 3 gpm pump, the pump runs continuously, and some of the water recirculates. there is an adjustment for the bypass pressure,, and I have had to adjust that, when a pump would not shut off.
That does make the pump act differently than the old one.
I wouldn't worry about the pressure.