It depends on how modulation is implemented. When there is a carrier with fixed power which is mixed with DSB modulated signal, then the power sum will be increased. But if it uses fixed power which is shared between carrier and DSB signal, then the power sum will be fixed and carrier power will be decreased for the amount of power which is transmitted with DSB signal, but this is not classic AM.
Since you have the power sum drop, maybe your power meter is not able to measure power properly when waveform is not pure sine (which happens when you put modulation signal).