Hmmm after discussing your problem with a couple of my colleagues, I realised there is a difference between PicoScope 5 and 6 in streaming mode, i.e. 200ms or slower for PicoScope 5 the device would do 1kS/s however in PicoScope 6 it goes much quicker! Now taking this into consideration it may be that the it is drawing more current.
What I would try to do in PicoScope 6 if you are in streaming mode (200 ms or slower ) is to reduce the number of samples for example say you choose 500ms choose 10kS or less.
I would also be interested in your settings, i.e. your time base, number of channels and number of samples.