If I re-scale my input range for differential input to ADC, can I double resolution?
When you talk about the resolution of an ADC, you are talking about the number of bits actually. So a 12 bit ADC is a 12 bit ADC and unless otherwise stated in the datasheet, it will remain a 12 bit ADC whatever your input range is.

However, by doubling the input range, you are actually doubling the step size of the ADC.

I'm not sure if this is a good or bad thing for you because this really depends