Usually you have to connect the accelerometer to the power supply (or signal conditioner), turn it on and wait few seconds to minutes for signal to settle. Then you're ready to proceed your measurements, depending on the system's time constant. A signal conditioner's constant current line drive (CCLD, also called ICP/IEPE/DeltaTron...) setting of 4 mA is a typical value. Some signal conditioners work as "Uni Gain", a gain factor close to 1.000 or gains in 10 dB steps (1, 3.14, 10, 31.4, 100 etc.).
The signal noise of +/- 10 g is not usual. Check the connections and signal conditioner settings (maybe the CCLD is set to 0 mA?). Check the integrity of your measuring chain (accelerometer, cable and signal conditioner) with proper test gear (e. g. an oscilloscope).
The offset you're describing during the switching shouldn't occur, unless you're resetting your signal conditioner (e. g. turning off and on) on each measurement.