I replaced my TPMS sensors, and I finally have working sensors, but the registered PSI number is about 3 PSI low on three accurate tire gauges. Is this normal?

Your vehicle dashboard gives you an accurate pressure reading while your shop gauge displays a relative pressure. A conventional shop gauge may display different results based on environmental factors like altitude and weather. The TPMS Pressure Gauge will always display the exact PSI it receives from the sensor inside the tire.