Why does changing the PGA setting from, say, 8 to 32 increase the current consumption of the circuit?

The PGA on a SigmaDelta ADC like the one on C8051F353 is usually implemented with CMOS switched capacitors. Essentially, there is a capacitor of a certain size that is being switched in and out at a certain frequency. This creates a variable resistor that can be scaled with frequency or capacitor size, relative to another reference capacitor in the design. The variable resistor is how we adjust the gain of the PGA. 

For certain gain settings, we scale the capacitor size and the for higher gain settings (over 8x), we adjust the frequency of the switched cap. The extra supply current is probably clock current going to those switched capacitors, and some from the switches themselves, being switched at a higher frequency. 

I am trying to use ADC on the C8051F35x and am having problems figuring out how the offset DAC works.

First of all, I am not sure if the offset DAC is added to or subtracted from the AIN+ and AIN- inputs such that it has the effect of level-shifting, is applied to only one input, or even if its doing the opposite on both inputs (half the value subtracted from AIN+ and half value added to AIN-, which would cancel any offset completely.

Also, application note AN217, which has the same block diagram as the 'F35x datasheet...

...shows that the offset should be applied before the gain, but in my tests it seems to be done afterwards.

From what I can tell, the offset voltage is substracted after the amplification. Is this expected behavior, or am I doing something wrong?

As the customer has noted, there appears to be a notable difference in how the offset DAC behaves relative to the PGA, and, in this case, the customer is correct: the offset DAC operates after the PGA. So, for starters, the block diagram above should be modified as follows:

Furthermore, consistent with the customer's experiments, the ODAC output is subtracted from AIN+ and added to AIN- as a signed value, where bit 7 and bits [6:0] of the ADC0DAC register represent the sign value (0 = positive, 1 = negative) and the magnitude, respectively.